To make the tart dough, cream the room temperature butter and sugar together using a stand mixer fitted with the paddle attachment, until light and fluffy and pale yellow.
Add the eggs one at a time and beat until fully mixed.
Add the flour and sugar-egg mixture, and mix until it forms a dough.
Gather the dough into a ball and chill.
Dust the work surface with flour, and roll out dough with a rolling pin and cut out into tart shells. Press dough into a tart pan. Prick tart shells with a fork before baking.
Preheat the oven to 350, and position rack in the center of oven. Bake the empty tarts for 7-8 minutes, until the tartlet cases are lightly golden.
To make the filling, melt butter in a heavy small saucepan over medium high heat, and cook butter until the butter is a deep nutty brown and there are brown flecks on the bottom of the pan. The browned butter will foam up and then settle, and have a nutty aroma.
Whisk sugar, eggs, vanilla, and salt together in a bowl. Whisk the flour in gently, then pour in the browned melted butter in a steady stream continuing to whisk constantly.
Place 2 -3 raspberries in each tart shell, and pour filling over top.
Bake the tarts at 350 for 10 minutes until they are golden brown and the filling is set.
Let cool in the tart pan on a wire rack. Remove from the pans, and dust with powdered sugar before serving.
Makes 20-22 tarts. Best eaten the day they are made, but leftovers can be stored in an airtight container.