Melt the butter in a small nonstick skillet over low heat. Meanwhile, beat the eggs and a pinch each of the salt and pepper with a fork in a medium bowl until well incorporated and air has been whipped into the eggs, about 1 minute.
Add the eggs to the skillet and stir constantly with a rubber spatula until the egg thickens, 1 to 2 minutes. Remove from the heat and continue to stir continuously until the eggs have a custard-like consistency, 1 to 2 minutes.
Place the soft scrambled eggs on top of the toast and sprinkle generously with the Parmigiano-Reggiano.