Step 1Heat the oven to 375 degrees. To make the crust, combine the flour, ground almonds, 3 tablespoons sugar and the salt in a bowl. Cut one-half cup cold butter into small pieces and work it into the dough with your fingers or a pastry cutter until the dough is crumbly and evenly combined but not pasty.
Step 2Combine the egg yolk, one-half teaspoon vanilla and 2 to 3 tablespoons whipping cream. Make a well in the center of the flour mixture and pour the egg yolk mixture in. Use a fork to quickly stir until the mixture can be formed into a ball.
Step 3Gather the dough into a ball and knead several times to blend the ingredients. Form into a ball again; wrap it in plastic wrap and chill 30 minutes.
Step 4Roll the dough out on a well-floured surface to about a three-eighths-inch thickness. Lift the dough into a 9-inch tart pan and gently press it into the bottom and up against the sides to the top edge of the pan; remove any excess dough. If you have any breaks in the crust, pat the broken area back together. Chill for 30 minutes.
Step 5Line the pan with foil, then fill halfway with pie weights. Bake the tart shell for 15 minutes. Remove it from the oven and lift off the foil and pie weights. Prick the bottom of the tart shell with a fork and return the crust to the oven. Bake until lightly browned, about 10 minutes. Remove the tart shell from the oven and cool on a wire rack. Reduce the heat to 350 degrees.
Step 6Arrange the apricot halves pitted-side-up in the tart shell.
Step 7Combine the remaining whipping cream, the remaining vanilla, the eggs and one-fourth cup of the sugar. Gently pour this custard over and around the apricots.
Step 8Dot the tops of the apricots with the remaining 2 tablespoons butter. Sprinkle with the remaining 2 tablespoons sugar.
Step 9Bake until the custard tests done in the center, about 35 to 40 minutes. Place the tart under the broiler until the top is browned, about 30 seconds or less.
Step 10Remove the tart from the oven and cool. Serve warm or chilled.