Heat the oven to 400°F (200°C) and butter a 3-quart baking dish. Whisk flour, cornmeal, sugar, baking powder, and salt in a medium bowl and set aside.
Melt the butter in a light-colored saucepan over medium heat for 4–6 minutes, swirling, until chestnut-brown specks appear and the aroma turns nutty. Pour into a cup and cool for 10 minutes.
Whisk both sugars, cornstarch, lemon juice, lemon zest, vanilla, cinnamon, cardamom, and salt in a large bowl. Fold in the peach slices until no white patches remain. Spread in the prepared baking dish and bake uncovered for 18–20 minutes, until bubbles appear around several edges.
While the fruit bakes, whisk the cooled browned butter and its solids into the cold buttermilk. Fold the buttermilk mixture into the dry ingredients just until the flour disappears — the dough should look rough.
Drop 10 dough mounds over the bubbling fruit, leaving open channels between them. Sprinkle with 1 teaspoon sugar. Bake at 400°F for 24–28 minutes, until the tops are deep golden, the filling bubbles through the gaps, and a center biscuit reaches about 200°F. Cover loosely with foil if tops brown before the center is done.
Rest the cobbler for 30 minutes before serving so the sauce thickens and the biscuit bottoms finish setting.