In a medium saucepan, heat the butter and olive oil over medium heat until the butter melts.
Add the flour and cook, stirring constantly, until the mixture thickens and becomes smooth, which takes about 3 minutes.
Stir in the grated zucchini and cook, stirring occasionally, for an additional 10 minutes.
Spread the mixture onto a dinner plate and allow it to cool, then refrigerate it until it's firm enough to handle.
Shape the zucchini mixture into 1-inch balls.
Place the beaten eggs and bread crumbs in separate shallow bowls.
Heat at least 1 inch of olive oil in a large skillet over medium-high heat, or, if available, use a deep fryer set at 365°F.
The oil should be hot enough that a cube of bread turns light brown in 60 seconds.
Dip each zucchini croquette in the beaten egg, then dredge it in the bread crumbs, and carefully place it in the hot oil.
Fry the croquettes, turning them several times, until they become golden.
Use a slotted spoon to transfer the fried zucchini croquettes to paper towels to drain any excess oil.