Turn on the heat to medium heat and whisk all the mixture. Then add soy sauce and bring it to a boil.
Once boiling, reduce heat to a gentle simmer (you should see bubbles around the edges) and continue simmering for 10-15 minutes (or 20 minutes if you're making the full amount). The liquid should reduce to roughly ⅓ of the original amount (The mark on a chopstick inserted in the sauce will give you a rough estimate).
Toward the end of cooking, you will see more bubbles. When the sauce thickens (check with the chopstick again to see if the sauce is ⅓ of the original amount), remove from the heat. As it cools, the sauce will thicken more. If you feel you've reduced too much, add a bit of water to loosen up and cook slightly more to the right consistency. You can store the sauce in an airtight jar and keep in the refrigerator for up to 3 months.
Preheat the broiler* to High (550ºF/288ºC) with a rack placed about 8" (20 cm) away from the top heating element (in the center of the oven) for 5 minutes. *Broiler setting: Low (450ºF/232ºC), Medium (500ºF/260ºC), and High (550ºF/288ºC). I usually use medium (6" away) or high (8" away). When broiling, you don't control the temperature in the oven; instead, you control the distance between the broiler and the surface of the food. It's similar to using hotter and cooler zones on your grill.
Line a baking sheet with foil for easy cleaning and brush/spray the foil with oil. Cut the unagi in half (or thirds, depending on the serving bowl size to fit the unagi fillet).
Place it on the foil, skin side down. Broil the unagi until the surface is blistered a bit, about 5-7 minutes. No need to flip.
Open the oven and brush the unagi with the sauce. Then broil again for 30-60 seconds until you see the sauce bubbling on top of unagi.
Preheat the oven to 425°F/218ºC with a rack placed in the middle and bake the unagi on parchment paper until the surface is blistered a bit, about 10-12 minutes. No need to flip. Brush the unagi with the sauce and bake again for 30-60 seconds until you see the sauce bubbling on top of unagi.
To Pan Fry
Wrap the unagi in foil (similar to this recipe) and reheat on low heat for 5-8 minutes. You won't get nice blisters/chars if you use this method.
Serve rice in a bowl and pour or brush unagi Sauce on the rice.
Serve unagi on top of rice and pour/brush more unagi sauce. Since I had a kinome (bud of Japanese sansho pepper tree), I garnish it on top (optional). You can also sprinkle Japanese sansho pepper at the table. Serve immediately.
To serve in the jubako (Japanese lacquered box), it looks nice to serve 2 pieces (the entire fillet), overlapping each other.
You can keep the leftovers in an airtight container and store in the refrigerator for 3 days or in the freezer for 2 weeks.