Make the filling: Preheat the oven to 350°F. In a saucepan, melt the stick of butter over medium-low heat, and let it go for 6 to 8 minutes, stirring occasionally, until the butter is fragrant and has a nutty aroma. The butter will foam and turn a rich caramel color. Remove from the heat and set the browned butter aside.
In a large bowl, combine the chopped apples, brown sugar, lemon juice, cornstarch, cinnamon, almond extract, and kosher salt.
Pour the brown butter across the bottom of a 7 x 11-inch baking dish. Then, evenly distribute the apples over the butter in the pan.
Make the cobbler topping: In a medium bowl, whisk together the flour, granulated sugar, baking powder, and kosher salt. Add the buttermilk and melted butter and stir using a spatula until all the ingredients are combined.
Pour or spoon the batter over the top of the apples and gently spread to cover them.
Bake until the filling begins to bubble around the edges and the cobbler topping is very lightly browned, 35 to 40 minutes.
For serving: Let rest for 10 minutes before scooping out the cobbler. Serve with ice cream or whipped cream and a sprinkle of flaky sea salt.