Place the unpeeled potatoes in cold water and boil until cooked through.
Wash, dry, and slice the trombetta zucchini into thin rounds.
Melt 20 g of butter with 2 tablespoons of olive oil in a pan over low heat.
If using shallots, finely chop them and cook in the butter and oil without letting them brown before adding the zucchini.
Add the zucchini rounds, season with a pinch of salt, freshly ground pepper, and nutmeg, and cook for a few minutes.
Continue to stir-fry over medium heat until cooked through, about 20 minutes.
Prepare the potato purée:
Drain, peel, and mash the hot potatoes using a food mill with a medium disc into a bowl.
Mix in the remaining butter, the shredded Emmental, the egg yolks, the grated Parmigiano, a pinch of nutmeg, and salt. Stir vigorously after each addition.