Crumble the matzah into a large sieve placed over a mixing bowl to catch the crumbs. Hold the sieve under running cold water until the matzah is moist and softened but not completely disintegrated, about 15 seconds. Transfer to the bowl with crumbs, then add the eggs and salt and mix gently with a fork.

Heat the butter in a 10- to 12-inch skillet over medium-high heat until foam subsides. Add the matzah mixture and cook, stirring constantly, until the eggs are scrambled and the matzah has begun to crisp, about 3 minutes.