Make the crust: pulse the almonds in a food processor until finely chopped, 30 to 40 seconds. Add the flour, butter, sugar, and salt. Pulse until the dough starts to come together, 1 to 2 minutes.
Press the dough into a 9-inch tart pan. Refrigerate until firm, at least 30 minutes and up to 2 days.
Make the filling: heat oven to 375° F with the rack in the lowest position.
Pulse the almonds in a food processor until finely chopped, 30 to 40 seconds. Add the butter, sugar, eggs, flour, vanilla, and salt. Process until the mixture is smooth, 1 to 2 minutes.
Place the chilled crust on a rimmed baking sheet. Pour the filling into the crust and smooth with an offset spatula. Top with the figs, cut-side up.
Bake until the crust and custard are golden brown, 30 to 40 minutes.
Combine the apricot jam with 1 tablespoon hot water. Brush top of tart with the apricot glaze.
Serve warm or at room temperature.