Puree ramps and olive oil in blender until fully combined—the oil should look green. Season with lemon juice and zest, salt, and pepper.
Trim asparagus and brush with olive oil. Roast asparagus over grill or griddle pan until tender, about 2 minutes per side.
Remove asparagus from heat and cut into bite-sized slices. Arrange asparagus on a plate. Top with burrata or mozzarella cheese, and drizzle with the ramp dressing.