Pulse the flour, powdered sugar, salt, and cold butter in a food processor 10 to 12 times, until the mixture looks like coarse sand with a few pea-size bits. Add the egg yolk and ice water and pulse until the dough clumps when pinched. Press it into a disk, wrap it, and chill for 45 minutes.

Heat the oven to 180 C / 350 F. Roll the dough on a floured surface to 3 mm thick, about a 30 cm round. Lift it into a 24 cm / 9.5-inch tart pan with a removable bottom, press it into the corners, trim the overhang, and prick the base with a fork. Freeze the lined pan for 15 minutes.

Line the shell with parchment, fill it with pie weights, and set the pan on a 33 x 46 cm / 13 x 18 inch baking sheet. Bake for 20 minutes, then lift out the parchment and weights and bake 5 minutes more, until the base looks dry and pale gold. Leave the oven on.

While the shell cools slightly, beat the softened butter and sugar in a 3-quart / 2.8 L bowl with a hand mixer for 2 to 3 minutes, until pale and fluffy. Beat in the eggs one at a time. Stir in the almond flour, 1 tbsp flour, vanilla, almond extract, and salt until smooth.

Pat the cut sides of the plum halves dry with a paper towel. Spread the frangipane into the shell in an even layer about 1.5 cm thick. Press the plums in cut side up, about a third of the way down, and scatter the sliced almonds over the top.

Bake on the baking sheet at 180 C / 350 F for 35 minutes, until the filling is deep golden, puffed around the plums, and springs back when pressed gently. A skewer inserted in the frangipane should come out clean.

Warm the jam and water in a microwave-safe bowl for 20 to 30 seconds, until pourable, then brush it over the hot tart. Cool the tart in the pan for 30 minutes, then push the base up to release it. Serve just warm or at room temperature.