Position rack in center of oven and preheat to 375°F. Using rubber spatula or fork, mix melted butter, sugar, and vanilla in medium bowl. Add flour and salt and stir until incorporated. Transfer dough to 9-inch-diameter tart pan with removable bottom. Using fingertips, press dough evenly onto sides and bottom of pan.
Bake crust until golden, about 18 minutes (crust will puff slightly while baking). Transfer crust to rack and cool in pan. Maintain oven temperature.
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. DO AHEAD: Can be made 1 day ahead. Cover and store at room temperature.
Remove tart pan sides. Place tart on platter. Cut into wedges and serve.
When browning butter, use a saucepan with a light-colored bottom so that you can gauge the color of the butter.