Preheat oven to 425ºF. Toss together first 4 ingredients in a large bowl; divide evenly between 2 rimmed baking sheets. Bake in preheated oven until golden brown, about 20 minutes. Reduce oven heat to 350°F.
Spread cornbread cubes evenly on a baking sheet; bake at 350°F until browned and crispy, about 15 minutes.
Cook butter, stirring constantly, in a medium skillet over medium until foaming. Add shallot and thyme; cook, stirring often, 1 minute. Drizzle butter mixture over toasted cornbread. Arrange Brussels sprouts in a serving dish; top with cornbread mixture.