On a lightly floured surface, roll Pâte Sablée into a 12-inch circle (about ⅛ inch thick). Carefully transfer to a 9-inch fluted round removable-bottom tart pan, lightly pressing into bottom and up sides. Using a paring knife, trim excess dough. Using a fork, evenly dock bottom of dough. Refrigerate until firm, at least 30 minutes.

Preheat oven to 325°F (170°C).

Top dough with a piece of parchment paper, letting ends extend over edges of pan. Add pie weights.

Bake until edges are set, about 15 minutes. Carefully remove parchment and weights. Bake until surface is dry and lightly browned, about 10 minutes more. Let cool completely in pan. Increase oven temperature to 375°F (190°C).

In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugar at medium speed until light and creamy, 2 to 3 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition and stopping to scrape sides of bowl. With mixer on low speed, gradually add flours, beating until just combined. Beat in vanilla, salt, and almond extract. Spread frangipane into cooled prepared crust. Press nectarine slices and blackberries into frangipane as desired.

Bake until frangipane is set and golden across surface, about 30 minutes, covering with foil during final 10 minutes of baking to prevent excess browning. Let cool in pan for at least 10 minutes. Remove from pan, and place on a wire rack.

In a small microwave-safe bowl, heat jam on high in 30-second intervals until soft and fluid. Lightly brush jam onto fruit in tart. Let cool completely before serving. Refrigerate in an airtight container for up to 3 days.