Preheat the oven to 200 degrees. Heat a griddle over medium. In a medium bowl, beat the eggs with the cheese and lemon juice; season with salt and pepper.
Drizzle the griddle with EVOO, then melt a dab of butter into the oil. Working in batches if needed, coat the bread in the egg mixture and cook until golden, 3 minutes per side. Transfer the toasts to a plate and keep warm in the oven.
Increase the heat to medium-high. In a large bowl, combine the chicken, parsley, capers, lemon zest, shallot, garlic, salt and pepper. Form the mixture into 4 patties. Drizzle the griddle with the EVOO, add the patties, and cook until browned and cooked through, 6 to 7 minutes per side. Drizzle the burgers with the lemon juice; divide among 4 toasts.
Add the spinach to the griddle; toss until wilted, then season. Pile the spinach on the burgers and top with the remaining toasts.