Preheat the oven to 375°F.

In a small pot or skillet, melt the butter over medium-high heat. Whisking occasionally, scrape up the browned bits as they form and continue to cook the butter until it smells like popcorn, 3–5 minutes. Remove from the heat and let cool slightly.

In a medium bowl, mix the flour, polenta, parmesan, salt, and sugar. Add 2 tablespoons water, plenty of black pepper, and pour the butter over. Using your hands, mix until well combined. The dough will be slightly sticky, but that’s okay.

Press the dough into a 9-inch tart pan or regular pie plate, making sure the sides are evenly packed, then press a nice, even layer on the bottom. Prick all over with a fork or tip of a knife.

Bake until golden brown on the sides and, most important, on the bottom, 20–25 minutes.

Set the crust aside and leave the oven on.

Arrange the tomato and garlic slices onto the parbaked shell in a nice, even layer, seasoning with pinches of salt, pepper, and red pepper flakes as you go. Top with capers (if using), then drizzle everything with 2 tablespoons olive oil, seasoning again with salt and pepper.

Place the tart pan on a sheet pan, return to the oven, and bake until the tomatoes are jammy and starting to brown and caramelize a touch on top, 55–60 minutes.

Remove the tart from the oven, splash with the vinegar, give it another drizzle of olive oil and let cool slightly before slicing. Serve with grated/shaved parmesan and some fresh herbs on top, if you’d like.