In a food processor, pulse the kale, basil, olive oil, minced garlic, grated Parmesan, lemon juice, and a pinch of salt and pepper until smooth but still slightly textured. Adjust seasoning if needed.

In a large bowl, whisk together the eggs, Greek yogurt, mirin, ginger powder, Monterey Jack cheese, Gruyere cheese, and a little salt and black pepper until glossy and uniform.

Heat a large nonstick skillet over medium heat and lightly butter or oil it. Pour roughly 1/4 of the egg mixture to form a pancake-shaped patty. Cook until the bottom is set and lightly golden, then gently flip and cook until just set through but still tender. Transfer to a plate and keep warm.

Preheat the oven to 350°F. Split the ciabatta rolls and arrange the bottoms on a baking sheet. Top each with caramelized onions and shredded mozzarella. Toast for 5–8 minutes until cheese is melted and edges are crisp.

Spread a generous layer of kale pesto on each toasted roll bottom. Place an egg patty on top, add more mozzarella if desired, cap with the roll top, and press lightly.