Whisk the eggs in a bowl until they are frothy, then whisk in the salt.
Heat the milk to 110° in a microwave or on top of the stove and stir in the yeast and sugar.
Combine the eggs and the milk mixture in the bowl of an electric food mixer equipped with a dough hook. Stir briefly to combine, then with the mixer set on slow, gradually add 3¼ cups of the flour. Increase the speed to medium and process for 10 minutes. The dough should cling to the dough hook and be fairly elastic. If the dough is soft and sticky, add the remaining ¼ cup flour. Reduce the speed to medium slow and continue to process for 5 more minutes, or until the dough is smooth, very elastic, and clinging to the dough hook.
Flour your hands well, then gather the dough into a ball. Place the dough in a heavy mixing bowl (a thick porcelain bowl is best) that has been lightly sprayed with vegetable spray. Very lightly spray the top of the dough, then put the dough in a warm, draft-free place. (We use the top of our stove, with the oven set at 300°. Or, if you have a gas oven with a pilot light, and you're not using the oven for anything else, put the dough in the unlighted oven, for the pilot light alone generates just enough warmth to encourage rising, and with the door closed, the oven is completely draft-free.) Cover the bowl with a kitchen towel and let the dough rise until doubled in volume, about 1 hour. Push the top down very gently-it will collapse easily.
On a floured surface, roll out a piece of dough about the size of a baseball into a circle about 6 to 8 inches across. Use plenty of flour and turn as you roll. If the round starts to stick, add more flour. Brush both sides of the dough round lightly with your choice of the seasoned butters (recipes follow) and set aside on a piece of baker's parchment paper. Let rise until doubled in bulk.
Preheat a 9- or 10-inch skillet over very high heat for 4 minutes, then reduce the heat to medium-high while frying the bread (a nonstick electric skillet set at 350° would be ideal). Fry the rounds, turning them several times, until crisp and nicely browned, about 3 to 5 minutes in all. Keep the first rounds warm while you fry the others, so you can serve everyone at the same time.