• Gather all the ingredients. Preheat the oven to 350ºF (180ºC). For a convection oven, reduce the cooking temperature by 25ºF (15ºC). Set the oven rack to the upper middle position for a standard oven or the middle position for a countertop oven.
  

To Prepare the Pan Pudding

• With ½ Tbsp unsalted butter, grease all sides of 2 individual gratin baking dishes. The dishes I used here measure 7 x 4½ x 1¼ inches (18 x 11.5 x 3 cm) and hold about 12 fluid oz (360 ml) each. For the best outcome, use dishes of a similar size or switch to 1 baking dish that‘s 8 x 8 inches (20 x 20 cm). Make sure your dishes are big enough to allow the bread plenty of wiggle room, as the bread pieces will expand in the oven to about 1½ times their original size. Tip: Your Pan Pudding will brown more easily if you use a shallow baking dish with low sides, which allow the hot air to circulate better.
  
• Cut 1 slice shokupan (Japanese milk bread) that‘s 2 cm thick into 16 square pieces. Place the squares in the baking dish(es), spacing them evenly throughout. Make sure there‘s plenty of room between the pieces for the bread to expand while baking; do not stuff the dish(es) with too much bread. Tip: Arrange the pieces so the brown crust is dispersed evenly among the white bread pieces. You want the final look to have brown spots here and there, and not just in one area.
  
• In a medium bowl, crack 2 large eggs (50 g each w/o shell) and beat until blended (I use a flat whisk and love it).
  
• Add ¼ cup sugar and ⅛ tsp Diamond Crystal kosher salt, and beat well until the sugar is completely dissolved. You want to make sure all the sugar goes into the custard and is not left in the bowl when you pour.
  
• Next, add ½ tsp pure vanilla extract and gradually whisk in 1½ cups whole milk. Blend well together.
  
• Slowly pour half of the custard mixture in a thin stream on top of the bread squares. Pause for a minute or two to allow the bread pieces to absorb some of the liquid. Then, pour in the other half of the mixture. It‘s okay if the bread pieces are floating. The custard will cook through in the oven. Tip: You might think that there is not enough bread in the dish, but this bread-to-custard ratio is key to the melt-in-your-mouth texture we want. Do not adjust by adding more bread or custard.
  

To Bake

• Bake the bread pudding at 350ºF (18ºC) for 30–40 minutes, or until the pudding rises and puffs up and the top of the bread is slightly golden. It‘s done when an inserted toothpick comes out clean and the internal temperature reaches 160ºF (71ºC). When I used my countertop Anova Precision Oven, it only took 20 minutes to bake, so please adjust the baking time based on your oven. When the bread pudding starts to puff up and rise but is not browned yet, make the caramel sauce (it takes 5 minutes).
  
• When the bread pudding is nice and golden, remove it from the oven. The bread pudding will deflate and fall once out of the oven, and this is normal. Tip: If your bread pudding is slow to brown on top, you can switch your oven to the Broil setting for 1–3 minutes until the bread is golden brown.
  

To Make the Caramel Sauce

• Before You Start: It takes about 5 minutes to make the caramel sauce and it requires your full attention while making it. If you are worried about your pan pudding and want to watch it closely, you can make the sauce after you remove the bread pudding from the oven. With more experience, you can time it better when you make it again. If you make the caramel sauce too early, it may solidify. Ideally, you want to finish both the bread pudding and the caramel sauce at the same time.
• In a heavy-bottomed small saucepan, combine 2 Tbsp sugar and 1 Tbsp water (at room temperature). Avoid a nonstick pot because it‘s too thin and the inner pot color makes it hard to see the caramel color. Turn on the heat to medium low and let the sugar dissolve completely, without stirring. You can tilt the pot occasionally, but otherwise don‘t touch it until the sugar turns liquid. In the meantime, prepare 1 Tbsp hot water. Have it ready to use in 3–4 minutes when the sauce is done.
  
• When the sugar has dissolved, increase the heat to medium high. The hot mixture will start to bubble. You can swirl the saucepan once in a while to mix it, but do not use a utensil because it will cause the sugar to crystallize. When the sugar starts to caramelize, you will see lots of small bubbles. It will slowly turn brown on the edges of the pan.
  
• Swirl the pot to incorporate the dark-colored caramelized sugar into the light-colored sugar. Soon, the bubbles will become larger as the sugar mixture caramelizes more and turns an amber color. The caramelized sugar will continue to turn darker and darker. Keep swirling the pot and watch it closely. 
  
• After the caramel sauce starts smoking, wait a tiny bit longer until it reaches an optimal medium to dark amber caramel color. Then, turn off the heat. Immediately pour a little bit of the measured hot water onto a spatula into the caramel pot (so your hand is not directly above the pot). 
  
• The mixture will boil vigorously and sputter, so be careful not to let it splatter on you. Once this has subsided, slowly add the rest of the hot water. Swirl the saucepan to combine.
  

To Serve

• Drizzle the caramel sauce over the hot bread pudding. Serve immediately with a spoon. Tip: Do you have some caramel stuck to the pot? Add milk and heat up while stirring. My kids enjoy caramel milk.
  

To Store

• You can cover it with plastic and store in the refrigerator for up to 2–3 days. I recommend consuming it sooner than later for the best taste.