Make the crust:
- Position rack in center of oven and preheat to 375°. 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. Prick all over with a fork.
- Bake crust until barely golden. Check at 10 minutes. (It may take a total of 15 to 18 minutes, but you only want the barest of color — it will brown with further cooking). Crust will puff a little when it cooks; poke with a fork to deflate if the puffs are dramatic. Remove and cool.
Make the filling:
- Whisk sugar, eggs and salt in medium bowl to blend. Add flour and vanilla; whisk until smooth. Melt butter in heavy small saucepan over low heat. When melted, increase heat to medium and cook until the butter solids turns a deep, nutty brown (do not burn), stirring often, about 6 minutes. Gradually pour browned butter into the sugar-egg mixture, whisking until well blended.
- Pour cherries into crust. Carefully pour browned butter mixture evenly over the fruit. Place tart on a 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. The tart can be eaten slightly warm from the oven, or cover and store at room temperature for up to one day.
- Remove tart pan sides. Cut into wedges and serve with a dollop of whipped cream, crème fraiche or vanilla ice cream.