Heat the butter and oil in a medium saucepan over medium heat until the butter melts.

Add the flour and cook, stirring constantly, until the sauce is thickened and smooth, about 3 minutes.

Add the coarsely grated zucchini and cook, stirring occasionally, for about 10 minutes.

Spread the mixture out on a dinner plate and let it cool.

Refrigerate until firm enough to handle (about 1 hour).

Shape the zucchini mixture into 1-inch balls.

Place the eggs and bread crumbs in separate shallow bowls.

Heat at least 1 inch of oil in a large skillet over medium-high heat (or use a deep fryer set at 365°F) until it turns a cube of bread light brown in 60 seconds.

Dip each croquette in the beaten egg, dredge in the bread crumbs, and place in the hot oil.

Cook, turning several times, until golden.

Transfer the croquettes with a slotted spoon to paper towels and let them drain.

Serve hot.