Place two slices of bread on a plate or flat surface.
Cut the avocado in half and remove the pit. Scoop the contents into a small bowl and mash with a fork.
Spread the mashed avocado evenly onto each bread slice and season with salt and pepper.
Place two slices of mozzarella cheese on top of the avocado on one slice of bread, then add the tomato slices, followed by the third slice of cheese.
Put the second bread slice on top of the sandwich and spread 1 tablespoon of softened butter evenly over the top of the bread, coating to the edges.
Heat a small skillet over medium heat and place the sandwich, buttered side down, onto the skillet. Butter the slice of bread that is facing up. Cook the sandwich over medium heat for 3-4 minutes per side, or until the crust is golden and the cheese starts to melt.
Remove from heat and serve immediately.