Preheat the grill or broiler.
In a 12 to 14-inch saute pan, heat the 6 tablespoons olive oil and the garlic slices together over medium heat, until the garlic begins to soften. Add the cavolo all at once, stirring to keep the garlic from settling at the bottom. Cover the pan and allow the cabbage to cook until soft, about 8 to 10 minutes.
Meanwhile, grill or toast the bread. Rub 1 side of each toasted bread slice with raw garlic. Place 1 slice of bread in 4 shallow bowls and top each slice with a large spoonful of the cabbage mixture. Drizzle each bruschetta with 1 tablespoons extra-virgin olive oil and shave pecorino over each portion.