FOR TART SHELL: In a large bowl, whisk together flour, sugar and salt. Add butter and rub mixture between fingers until
it resembles coarse meal. Drizzle 1 tablespoon water and rum over mixture and sprinkle with zest; mix together to form a dough. Knead dough 3 to 4 times, flatten into a disc, wrap in plastic and chill for at least 1 hour.

Butter tart pan. On a lightly floured work surface, roll out dough to a 12-inch round. Fit dough into tart pan, pressing into edges. Indent edges with tines of a fork and prick bottom of dough all over with fork. Chill until firm, about 30 minutes.

Meanwhile, heat oven to 400º with rack in middle. Line tart pan with parchment paper, leaving at least a 1-inch overhang. Fill with pie weights or dried beans. Bake until edges are pale golden, about 10 minutes. Remove parchment and weights; continue to bake until bottom of crust is pale golden and edges are beginning to brown, about 5 minutes more. Cool shell completely on wire rack.

FOR FILLING: Reduce oven temperature to 350º. In a large bowl, whisk cheese until smooth. Add egg yolks, 3/4 cup sugar,
2 tablespoons amaretto and almonds. Whisk together to combine.

Spread filling into cooled tart shell and bake, rotating once halfway through, until just set in center and a cake tester inserted in center of tart comes out clean, about 45 minutes. Transfer to a wire rack.

In a large skillet, combine remaining 1/4 cup sugar and butter. Heat over medium heat until butter is melted and lightly golden, then reduce heat to low and add apricots, cut-side down, and remaining 2 tablespoons amaretto. Bring liquid to a simmer and cook apricots, turning once, until fruit is tender and glaze is thickened, about 4 minutes. Arrange apricots on tart, cut-side down, then drizzle with glaze. Let tart cool completely before serving.