1. Preheat your oven to 400°F (200°C). Line a baking dish with parchment paper or lightly brush with oil (I used an 8.5 × 6.5 inch / 22 × 16.5 cm Pyrex glass dish). 
2. Use a vegetable peeler (or a mandolin slicer) to slice the zucchini into long, thin ribbons. You should get about 30 thin slices.

3. Place the zucchini slices on a plate and toss them gently with the oil so they are lightly coated.
 
4. On a large plate, mix the breadcrumbs, oregano, and salt.

5. Place two zucchini strips side by side, overlapping them slightly along their length to form one wider strip. Press both sides into the breadcrumb mixture. Lay the zucchini flat on a cutting board. Add one slice of prosciutto and thin strips of cheese on top. Roll it up gently. Repeat with the remaining strips of zucchini, prosciutto, and cheese. 


6. Place the roll-ups snugly next to each other in the baking dish in a single layer.

7. Bake in the preheated oven for about 20 minutes, until the cheese is melted and the breadcrumbs are golden brown. Sprinkle Parmesan cheese on top. Serve warm and enjoy.