1. Preheat the oven to 375
2. In a small saucepan, melt the butter over medium heat, then cook, stirring frequently, until deep golden brown but not burnt, about 5 minutes. Remove from the heat and set aside.
3. In a medium bowl, combine ⅓ cup of the sugar, 1 ¾ cup of the flour, and the salt. Add ½ cup of the browned butter and the vanilla extract, then mix with your fingers until the mixture is crumbly and homogeneous, about 6 minutes.
4. Press the mixture into an even layer over the bottom and up the sides of a 9-inch tart pan with a removable bottom.
5. Use a fork to poke eight or nine holes all over the bottom of the tart shell and bake for 8 minutes, then rotate the pan 180 degrees and bake for 8 minutes more, until lightly golden. Remove the tart crust from the oven and let cool.
6. In a large bowl, whisk together the eggs, vanilla bean seeds, and remaining ½ cup sugar until combined, then whisk in the remaining browned butter and ¼ cup flour; set aside. It should look thin and drippy.
7. Arrange the raspberries and blackberries in the tart pan, pointy-side up, starting with one ring of blackberries along the outside edge, then filling in with smaller and smaller rings of raspberries and ending with a small cluster of blackberries in the center.
8. Carefully pour the custard over the berries, filling in the spaces between them as best you can. Tap the tart pan on the counter so the filling oozes between the berries. Return the tart to the oven and bake, rotating the pan halfway through the cooking time, until the filling has risen, the top is golden brown, and a toothpick inserted into the center of the tart comes out dry (not including berry juice!), 35 to 40 minutes. Remove from the oven and let cool for 10 minutes, then sprinkle with confectioners’ sugar, if desired. Slice and serve while slightly warm or at room temperature.