Brown the butter for the base: Add 9 tablespoons butter to a light-colored saucepan over medium heat. Cook, swirling, until the milk solids turn deep golden and smell nutty, 4 to 6 minutes. Scrape into a heatproof bowl and let cool to room temperature (it should be fluid, not hot).
Line a 9×13-inch metal baking pan with parchment, leaving overhang on the long sides for easy lifting. Position a rack in the lower-middle of the oven and preheat to 340°F (170°C).
Make the gooey filling first: In a large bowl, beat the softened cream cheese on medium speed until completely smooth, about 1 minute. Mix in the melted butter and sour cream until silky. Beat in the eggs and vanilla just to combine. Gradually add the powdered sugar on low speed, then increase to medium and beat until thick and glossy, 45 to 60 seconds. Set aside; bubbles will settle while you prepare the base.
Whisk the dry ingredients for the base in a separate bowl: flour, baking powder, and salt. In another bowl, whisk the cooled browned butter with granulated sugar, egg, egg yolk, vanilla, and lemon zest until smooth and slightly lightened, 30 seconds. Add the dry mixture and stir with a spatula just until no dry streaks remain; the dough will be thick.
Press the dough evenly into the prepared pan. Use a piece of parchment or damp fingertips to smooth the surface; lightly dock the crust with a fork in a few places. Par-bake for 7 minutes to set the top.
Give the filling a quick stir, pour it over the warm crust, and tap the pan firmly on the counter 2 to 3 times to release air pockets. Bake until the edges are puffed and lightly browned and the center has a gentle, 1- to 2-inch wobble, 28 to 33 minutes, rotating the pan once halfway. If browning quickly, tent loosely with foil for the last 5 minutes.
Cool the pan on a rack until just warm, about 45 minutes, then refrigerate uncovered until fully chilled and set, at least 2 hours. Lift out using the parchment, dust generously with powdered sugar, and cut into 20 squares with a hot, dry knife for clean edges.