Preheat an oven to 375 degrees Fahrenheit and position an oven rack in the bottom third of the oven.
Lightly dust a clean work surface with flour and roll chilled pastry dough into a circle with a 14-inch diameter. Fit dough circle into a 10-inch tart pan, being careful not to stretch out the dough too thin. Bring the dough up the sides of the pan and trim any excess dough. Transfer to the freezer and let chill, about 15 minutes.
To prepare the frangipane: In a food processor fitted with a metal blade, finely grind the almonds and set aside. In the bowl of a stand mixer, combine the butter and sugar and beat until light and fluffy, about 2 minutes. Add the ground almonds, egg, almond extract, flour and salt and beat until smooth. Spread evenly over the bottom of the chilled tart shell.
Place each apricot half, cut side down, around the edge of the tart, leaving a space between each half and place one half in center of tart. Transfer the tart to the oven to bake until the tart shell is golden brown and the frangipane is puffed and brown, 35 to 40 minutes.
To prepare Cointreau cream: While the tart is baking, place the cream and vanilla in a medium mixing bowl and beat with an electric mixer on medium-high speed until the mixture visibly thickens and begins to hold a shape. Beat in the Cointreau and set aside in the refrigerator until ready to use.
To prepare apricot glaze: While the tart is baking, melt apricot jam with water in a small saucepan over medium heat. After the tart is finished baking, use a silicone pastry brush to lightly coat the top of the tart with glaze. Cool tart to room temperature before slicing.
To serve: Slice tart into wedges and serve with Cointreau cream on the side.
Flaky Pie Dough
To prepare the dough: Place the flour and salt in a large mixing bowl. Using a pastry blender, two knives or your fingers, cut or rub the butter into the flour mixture until the butter pieces are the size of peas. Sprinkle ice water, 1 tablespoon at a time, over the flour mixture. Stir and fluff the flour mixture with a fork until shaggy clumps form. Pick up the large clumps and squeeze to test for moisture. If clumps hold together, transfer them to a work surface and continue adding tablespoons of ice water and stirring until all the dough passes the squeeze test.
Gather the clumps of dough together on a work surface. If the butter feels soft, cover the dough with plastic wrap and refrigerate at least 30 minutes before proceeding. If the butter still feels cold, knead the dough gently 4 or 5 times. If dough falls apart, return it to the mixing bowl and sprinkle teaspoons of ice water and mix with a fork until dough holds together when squeezed.
Flatten the dough into a disk about 6 inches wide, cover in plastic wrap and refrigerate for 30 minutes before rolling.