Melt the butter in a small pan over medium heat until the milk solids turn amber and smell nutty, then chill it until firm but pliable.
Heat the oven to 375°F and butter a 9-by-13-inch baking dish.
Toss the apples with the first brown sugar, 2 tablespoons flour, lemon juice, cinnamon and 1/4 teaspoon salt; spread in the dish.
Mix the remaining flour, remaining brown sugar and 1/2 teaspoon salt, then rub in the cooled brown butter to make crumbs.
Scatter the crumbs in an even layer over the apples.
Bake for 45 to 50 minutes until the apples are tender, the center bubbles and the crumbs are browned; rest 20 minutes.