Peel the potatoes and cook in a pot with salty water until tender. When the potatoes are done, drain them and purée them through a food mill and onto a work surface while they are still warm.

Add most of the flour to the potatoes and the egg, knead into a smooth ball. Since some potatoes absorb less flour than others, it is best not to add all the flour but little at a time. Stop adding the flour when the mixture has become soft and still lightly sticky.

Dust the work surface lightly with flour. Divide the potato mixture into 2 or more parts and shape each of them into a sausage-like roll about 1 inch thick. Slice the rolls into pieces 3/4 inch long.

To shape the gnocchi hold a dinner fork more or less parallel to the counter and with the concave side facing you. With the index finger of your other hand hold one of the cut pieces against the inside curve of the fork, just below the tips of the prongs. At the same time you are pressing the piece against the prongs, flip it away from the tips and in the direction of the fork's handle. Let it drop on the counter.

Set 4 quarts of water to boil in a broad pan (about 6 quarts capacity), add salt. Drop the first full batch of gnocchi, about 2 dozen in the boiling water. When they start floating to the surface retrieve them with a colander scoop or a large slotted spoon, and transfer them to a warm serving platter. Spread over them some of the pesto and the cheeses. Drop more gnocchi in the pot and repeat the whole operation.