Pulse flour, sugar, and salt in a food processor. Add cold butter and pulse to coarse sand with pea-sized bits. Add egg yolk and pulse until clumping, then add ice water 1 tablespoon at a time until dough holds together.
Shape dough into a disc, wrap in plastic, and refrigerate for 30 minutes.
Roll chilled dough into an 11-inch circle, ⅛ inch thick. Press into a 9-inch tart pan, trim excess, and prick the bottom with a fork. Freeze for 10 minutes while preheating oven to 375°F (190°C).
Line the chilled crust with parchment and fill with pie weights. Bake for 15 minutes, remove weights and parchment, then bake for another 5 minutes until dry and barely golden. Remove and reduce oven temperature to 350°F (175°C).
Beat softened butter and sugar until light and fluffy, about 2 minutes. Beat in almond flour, then add eggs one at a time. Mix in all-purpose flour, vanilla, and rum until smooth and thick.
Spread frangipane evenly in the blind-baked crust. Arrange peach slices in a circular, overlapping pattern. Sprinkle with granulated sugar.
Bake at 350°F (175°C) for 35 minutes, until frangipane is puffed and golden and peaches have softened. Cover crust edges with foil if browning too quickly.
Warm apricot jam with water until brushable. Brush warm glaze over peach slices. Cool in the pan for at least 30 minutes before removing the outer ring and slicing.