Preheat the oven to 350ºF. Liberally coat a 9x13 pan with baking spray or grease with butter.
For the batter, start by using a whisk to combine the milk, eggs, and vanilla in a liquid measuring cup or bowl. Set this aside.
1 ½ cups whole milk, 2 eggs, 2 teaspoon vanilla
In a large mixing bowl, mix together the flour, sugar, baking powder, and salt. Pour the wet ingredients on top of the dry ingredients, then use a rubber spatula to stir the batter until it is just barely combined.
3 cups all-purpose flour, 1 cup granulated sugar, 4 teaspoon baking powder, ¼ teaspoon kosher salt
Add 4 tablespoons of the butter you browned earlier on top of the batter. Continue to stir gently with the spatula until the batter is evenly combined.
4 tablespoon browned butter
Pour the batter into the prepared pan. Use a spoon to dollop the filling across the top of the batter (I do three even dollops in the center, then six more on each side to make five rows of three. See the post for a picture). Use a butter knife to gently swirl each dollop into the surrounding batter all the way to the edges.
Bake for 25-25 minutes, or until a toothpick inserted in the center of the pan comes out clean or with just a few moist crumbs. Allow to cool on a wire rack for at least 5 minutes while you make the icing.