For the tart shell:
1/8 tsp. baking powder
1¾ cups cake flour
5 oz. unsalted butter, softened
1/4 cup powdered sugar
1 large egg
For the frangipane filling:
12 oz. almond paste
8 oz. unsalted butter, softened
5 large eggs
1/4 cup cake flour
To finish:
2 Granny Smith apples, about 4 oz. each, peeled and thinly sliced
For the apricot glaze:
1/4 cup warm water
3/4 cup strained all-fruit apricot jam
Combine the baking powder and cake flour in a bowl.
Combine the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until smooth, one to three minutes. Add the egg and beat on low speed until incorporated. Add the cake flour mixture and beat just until incorporated. Remove the dough, pat into a disc, wrap in plastic wrap, and refrigerate it for one hour.
Preheat the oven to 350°F.
Roll the chilled dough out on a floured surface to an 11-inch circle approximately 1/4-inch thick. Fold the dough into a 10-inch tart pan and press it evenly out on the bottom and up the sides to the top of the pan. Line the tart shell with an 11-inch circle of parchment or heavy foil and fill it with baking beans or pie weights. Bake it for 15 minutes, or until partially baked, with a pale colored center and light golden brown edges. Cool to room temperature and remove the bean-filled parchment or foil.
For the frangipane filling:
Combine the almond paste and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until smooth, three to five minutes. Scrape the bowl well. Add the eggs one at a time, beating them in on low speed to incorporate and scraping the bowl after each edition. Add the cake flour and beat just until incorporated.
To finish:
Preheat the oven to 350°F.
Fill the cooled tart shell with frangipane filling until the top is level with the edge of the crust. Fan the apple slices, arranging them into a circle on the top and press them slightly into the frangipane. You may use as many apple slices as desired and change the pattern as well. Bake the tart for 20 to 30 minutes, or until the filling is golden brown and the center of the tart springs back when touched. Let the tart cool slightly, then brush it with the apricot glaze. Serve the tart with whipped cream or ice cream.
For the apricot glaze:
Combine the water and jam in a small saucepan and bring to a boil over medium-high heat. Remove from the heat and stir well. Use a pastry brush to brush it on the tart.
Covered, the tart will keep in the refrigerator for up to three days. Well wrapped, it can be frozen for up to a month. Bring it to room temperature and glaze just before serving.