Make a "partially baked" crust by following this step. You can make the tart crust ahead of time.
To Make the Almond Cream Filling (Frangipane)
Gather all the ingredients. Make sure to take out the butter and eggs from the refrigerator and keep at room temperature. If the eggs are too cold when you add to the mixture, they reduce the temperature of the butter, making emulsion more difficult. Adding the eggs all at once also inhibits emulsion, and results in that scrambled egg appearance. You can soak the cold eggs for 5 minutes in warm (not hot!) water, but it's best to take out both butter and eggs out from the fridge earlier.
In a large bowl, cream the softened unsalted butter.
Add confectioners’ sugar and mix well until the mixture resembles a coarse meal.
Add almond flour, all-purpose flour, and salt, and mix well.
Add eggs one at a time and incorporate into the mixture.
Add vanilla extract, almond extract, and rum.
Whisk until homogenized and smooth. The almond cream can be kept refrigerated for up to 3 to 5 days or frozen for weeks.
For the Assembly
Preheat the oven to 375ºF (190ºC). For a convection oven, reduce cooking temperature by 25ºF (15ºC). Melt the apricot jam and 1 Tbsp water in the microwave for 10 seconds until liquefied. Apply the jam on the base of the cooled pastry case to prevent soggy bottom. Reserve the jam for later. Fill the baked tart crust with almond cream, smoothing out to the edges with an offset spatula, and refrigerate for 30 minutes to set.
Meanwhile, peel the pears, cut each in half lengthwise, and scoop out cores.
Cut each half crosswise into ⅛ inch (3 mm) slices. Squeeze the lemon and rub the pears with lemon juice to prevent browning.
Take out the tart crust from the refrigerator. Gently press each pear half to fan out toward the stem/top side. Slide the knife under pears and transfer to the almond filling.
Arrange the top with points facing in and pears fanning out toward the center. Make sure to space pears out so they are not touching, as the filling will puff up and they need room to expand.
Bake the Tart
Bake the tart at 375ºF (190ºC) until golden brown or the bamboo skewer inserted into the center of filling comes out clean, about 40-45 minutes.
Remove the tart from the oven and let cool in the pan on the wire rack. Brush the pears with the leftover apricot jam. Once it’s cool, carefully unmold the tart.
Toast the almond slices in the frying pan (no oil) until golden, about 5 minutes.
Sprinkle confectioners’ sugar and toasted almond slices on top, if desired, and serve slightly warm. Cut into wedges and enjoy!
The tart can be stored in the refrigerator up to 3 days. Slightly re-heat the tart before serving.