Position rack in center of oven andpreheat to 375°F. Using rubber spatula orfork, mix melted butter, sugar, and vanillain medium bowl.
Add flour and salt andstir until incorporated.
Transfer dough to9-inch-diameter tart pan with removablebottom. Using fingertips, press dough evenlyonto sides and bottom of pan.
Bake crust until golden, about 18minutes (crust will puff slightly whilebaking).
Transfer crust to rack and cool inpan. Maintain oven temperature.
Whisk sugar, eggs, and salt inmedium bowl to blend.
Add flour and vanilla;whisk until smooth. Cook butter in heavysmall saucepan over medium heat until deepnutty brown (do not burn), stirring often,about 6 minutes. Immediately pour brownedbutter into glass measuring cup. Graduallywhisk browned butter into sugar-eggmixture; whisk until well blended.
Arrange raspberries, pointed side upand close together in concentric circles,in bottom of cooled crust. Carefully pourbrowned butter mixture evenly over berries.
Place tart on rimmed baking sheet.
Bake tartuntil filling is puffed and golden and testerinserted into center comes out clean, about40 minutes. Cool tart completely in pan onrack. DO AHEAD: Can be made 1 day ahead.Cover and store at room temperature.
Remove tart pan sides.
Place tart onplatter.
Cut into wedges and serve.
When browning butter,use a saucepan with a light-colored bottomso that you can gauge the color of the butter.