Trim the edge of the dough with the rolling pin. Prick the shell with a fork and chill the shell for 30 minutes.
Preheat the oven to 350 degrees and preheat the baking sheet.
In a bowl with the mixer cream the butter, add 1/4 cup plus 2 teaspoons of the sugar, and beat the mixture until fluffy. Add the egg, beating well, and beat in the almonds and the vanilla. Spread the frangipane mixture in the shell, top it with the apricot halves cut sides up, and bake the tart on the baking sheet in the lower third of the oven for 40 minutes, or until the filling is golden and set.
Preheat the broiler.
Sprinkle the tart with the remaining 2 teaspoons sugar and put it under the broiler about 6 inches from the heat for 2 to 3 minutes, or until the filling is browned. Brush the tart with the apricot glaze, let it cool, and remove the rim of the pan.
Recipe courtesy of Gourmet Magazine