Heat grill to medium-high. Brush both sides of bread slices with 1/2 cup oil. Grill until golden and charred in spots, about 1 minute each side. Season with salt. Dollop cheese among slices.
Toss asparagus with remaining 1 tablespoon oil; season with salt. Grill, turning occasionally, until crisp-tender, 5 to 7 minutes. Divide among bread slices. Top with sprouts; drizzle with oil. Season with salt and pepper flakes; serve.