Rinse and gently rub the avocado with water. Pat dry with paper towel. Cut the avocado in half. Remove the seed. Place half of the avocado on a chopping board flesh side up, using a paring knife, score the inside of the avocado into cubes. Scoop out the flesh. Drizzle the lemon juice over. Toss together with a big spoon. Season with salt and keep mixing until creamy and the avocado pieces are about pea-sized.
In a large nonstick skillet over medium heat, melt 1 tablespoon butter and wait until the foaming stop. Add 2 slices of bread side by side. Cook until light golden brown on the bottom side, about 2 minutes. Transfer to a chopping board with untoasted side up. Add another tablespoon butter and toast the other 2 slices of bread. Wipe the skillet clean with paper towels.
To assemble, place 2 slices cheese to cover the toasted side of the bread. Divide the avocado and add half on top of that slice of bread. Spread to make an even layer. Top with another 2 slices of cheese. Place another slice of bread toasted side down. Repeat with the other sandwich.
In the same skillet over medium-low heat, melt 1 tablespoon butter. Add both sandwiches and cook until deep golden brown, about 5 minutes. Remove the sandwiches and add another tablespoon butter. Carefully flip the sandwiches and cook until the other side is also deep golden brown, about 5 more minutes. Sprinkle with a small pinch of salt. Transfer to a chopping board. Cut in half. Serve immediately.