Place the broccoli and cauliflower in a large bowl. Toss with the beaten eggs to coat evenly. Place the Parmesan in a large tray or baking dish and dredge the vegetables in the cheese, pressing to coat evenly.
Pour olive oil in a large heavy frying pan until the olive oil is 0.5cm deep, about 120ml depending on the size of your pan. Warm the olive oil over medium-high heat. When the oil is hot, carefully add the Parmesan-coated vegetables. Do not overcrowd the pan. Cook in batches, if necessary. Let the vegetables cook until a crust forms, about three minutes per side. Turn the vegetables only when they easily release from the bottom of the pan. Drain on paper towels and season with salt.
Place the spinach in a large bowl. In a small bowl, whisk together the lemon zest, lemon juice, extra-virgin olive oil, salt, and pepper. Drizzle the spinach with the vinaigrette and toss to coat. Add the Parmesan coated vegetables.
Toss and serve.