Grill bread slices over medium high heat until golden brown on both sides. Rub one side of the bread with garlic; brush lightly with oil. Place in a single layer on a baking sheet.
Preheat broiler. Bring a large saucepan of water to a boil over high heat; add a generous pinch of salt. Add the asparagus and cook until tender, 3 to 5 minutes. Drain the asparagus and set aside.
Combine the Monterey Jack and Parmesan in a bowl. Sprinkle two-thirds of the cheese evenly over the ciabatta slices. Arrange 3 asparagus spears on top of each slice. Sprinkle the remaining cheese over the asparagus, dividing evenly. Season with salt and pepper.
Broil until the cheese has melted, 1 to 2 minutes. Remove from the oven and cut in half.
Serve warm.