Melt the 2 teaspoons butter in a sauté pan over medium heat. Crack the eggs and place them in the pan. Cook until the whites are set and barely crispy. Flip the eggs and cook for 30-45 seconds. Dish out and set aside. Spread equal amounts of mayonnaise on each of two slices of bread (mix the mayonnaise with basil if desired). Top with equal amounts of cheese, tomato, and avocado slices. Add one egg on top of each sandwich. Cover with second piece of bread. Melt half the remaining tablespoon of butter in the sauté pan over medium heat. Place the sandwiches in the pan. Place another, heavier pan on top. Cook for about 2 minutes. Remove the heavier pan, lift the sandwiches with a spatula and add the remaining butter to the pan. When the butter has melted, turn the sandwiches on the uncooked side, weight down with the heavier pan and cook for another minute or so or until second side is golden brown.
Ronnie Fein is a cookbook author, food writer and cooking teacher in Stamford. She is the author of The Modern Kosher Kitchen and Hip Kosher. Visit her food blog, Kitchen Vignettes, at www.ronniefein.com, friend on Facebook at RonnieVailFein, Twitter at @RonnieVFein, Instagram at RonnieVFein.