Place cauliflower in a skillet over medium heat; drizzle with olive oil. Cook and stir until light golden, 5 to 7 minutes. Pour cooked cauliflower into a blender with Parmesan cheese, lemon juice, and black pepper. Blend until the mixture resembles mashed potatoes, 10 to 15 seconds.
Toast bread, about 2 minutes, and spread light mayonnaise equally on each slice. Spread cauliflower mash equally over the toast. Place avocado slices over the cauliflower.