Preheat oven to 400°. Cook butter in a large saucepan over medium heat, swirling pan and stirring occasionally to prevent solids from sticking to bottom, until butter foams and then browns, 5–8 minutes. Remove from heat; add cornbread, salt, and 4 Tbsp. sugar and fold gently until cornbread is evenly coated.
Spread out on a rimmed baking sheet and bake until crumbs are golden brown, 10–15 minutes (crumbs will look wet but will crisp and dry out as they cool). Let cool.
Meanwhile, combine berries and remaining 2 Tbsp. sugar in a large bowl. Let berries sit, tossing occasionally and mashing a few with a spoon, until some juices accumulate, 15–20 minutes.
Divide ice cream among small bowls and spoon berries plus juices over top. Sprinkle with some of cornbread crumble.
Do Ahead: Crumble can be made 3 days ahead. Store airtight at room temperature.