In a small pot over medium heat, melt butter. Once melted, reduce heat to medium-low and continue to cook, watching closely and stirring occasionally, until butter simmers and starts to turn golden. Once simmering stops and butter is golden and smells nutty, about 5 minutes, remove from heat. Pour into a medium heatproof bowl and stir in vanilla. Let cool 15 minutes.
Preheat oven to 375º. In a large bowl, whisk flour, granulated sugar, baking powder, and salt. Add cooled brown butter and stir until a crumbly dough forms. Scrape dough into a 9" tart pan with a removable bottom. Using your fingers or back of a measuring cup, press dough in bottom and up sides of pan. Place pan on a small rimmed baking sheet and freeze 15 minutes.
Keeping pan on baking sheet and using a fork, poke bottom of tart shell all over. Line tart with parchment and fill with pie weights or dried beans.
Bake until edges are golden, about 20 minutes. Remove parchment and weights and continue to bake until bottom is golden and cooked through, about 5 minutes more. Transfer pan to a wire rack and let cool.