Heat the olive oil in a pan and gently sauté the chopped onion with a pinch of salt over low heat until it's soft and translucent.
Stir in the flour and cook for about 2 to 3 minutes, mixing constantly so it doesn't clump or burn.
Slowly pour in the warm milk while stirring, then cook until the mixture thickens into a creamy sauce.
Add salt, pepper, and the shredded cheese, stirring until the cheese melts completely into the sauce.
Mix in the diced smoked beef or turkey ham, then transfer the filling to a bowl and press plastic wrap directly onto the surface; let it cool to room temperature.
Slice the zucchinis lengthwise into thin strips (about 3 mm thick) using a mandoline or sharp knife until you have about 22 strips.
Blanch the zucchini slices in salted boiling water for 1 minute, then transfer to a bowl of cold water and pat dry with paper towels.
Spread a spoonful of the cooled filling onto each zucchini strip, then roll them up tightly.
Dredge each roll in flour, dip in beaten egg, then coat with breadcrumbs until fully covered.
Heat olive oil in a skillet and fry the roll ups until golden brown on all sides, turning as needed; drain on paper towels before serving hot.