This tart smells so good while baking that it's all I can do not to pull it out of the oven and go at it with a large spoon.
Preheat oven to 350. Beat together butter, 1/4 cup white sugar, brown sugar, and salt until fluffy. Add 3 eggs and 1 teaspoon vanilla and beat until smooth (may take several minutes to prevent curdling). Stir in flour until just combined.
Pour batter into tart pan and spread evenly with spatula, making sure to spread it up the sides of the pan. Batter will be sticky but should be thick enough to stay put.
In a separate bowl, thoroughly mix creme fraiche with remaining egg, 1/4 cup white sugar, and 1 teaspoon vanilla. Pour into tart pan, spreading to edges of crust batter.
Drop raspberries into filling, spacing as evenly as possible. Sprinkle with vanilla sugar and bake 50-60 minutes, until edges of crust are golden and filling is set. Remove from oven and dust with powdered sugar. Let cool completely before slicing.