In a small pan over medium heat, warm 2 tablespoons of the oil and the halved garlic slices, cut side down, until the garlic is very aromatic and just starting to turn golden. Remove and reserve the garlic. Add to the oil the breadcrumbs and 1/4 teaspoon salt. Cook, stirring frequently, until the breadcrumbs are crisp but not burning, about 5 minutes. Set aside.
Preheat the oven to 400°F. On a large rimmed baking sheet, toss together the brussels sprouts and remaining 2 tablespoons oil, then spread into 1 layer and roast until tender and caramelized, about 18 minutes.