Whisk sugar, eggs, and salt in medium bowl to blend. Add flour and vanilla; whisk until smooth. Cook butter in heavy small saucepan over medium heat until deep nutty brown (do not burn), stirring often, about 6 minutes. Immediately pour browned butter into glass measuring cup. Gradually whisk browned butter into sugar-egg mixture; whisk until well blended.
Arrange raspberries, pointed side up and close together in concentric circles, in bottom of cooled crust. Carefully pour browned butter mixture evenly over berries.
Place tart on rimmed baking sheet. Bake tart until filling is puffed and golden and tester inserted into center comes out clean, about 40 minutes. Cool tart completely in pan on rack.
Remove tart pan sides. Place tart on platter. Cut into wedges and serve.