Fill a tall saucepan with water and heat it with the vinegar until it starts to boil.
Swirl the liquid with a spatula to create a vortex in the center and poach the eggs, one at a time, for three minutes.
Drain the eggs with a slotted spoon and let them cool for 2 minutes in very cold water.
Dry them on absorbent paper and set them aside.
Heat the cream in a saucepan and reduce for about 10 minutes, adjusting with salt and pepper.
Finely chop the well-washed and dried mint and cut the sweet Provolone Valpadana DOP into 4 1 cm thick slices.
Grate the zest of the two lemons and set it aside.
Place 2 eggs in each cocotte, cover with cream, sprinkle with mint and add the cheese slices on top. Bake in grill mode, at 220°, about 4 minutes, just enough time to brown the sweet Provolone Valpadana DOP.
Remove from the oven, add the lemon zest and serve with the croutons.
