Preheat oven to 400°F

Melt butter in a large, oven safe skillet over medium heat

Once butter is melted, carefully and using an oven mitt, swirl the skillet so the bottom and sides are coated in butter (this prevents the egg mixture from sticking to the sides later)

Gradually add thinly sliced onions

Add 1/2 tsp salt, mix

Work with a wooden spoon to make sure all onions are evenly coated in butter/salt, about 2-3 minutes

Reduce heat to medium-low

Sauté for 10 minutes, tossing with a wooden spoon every 2-3 minutes

After 10 minutes, add the eggplant, fold into the onions

Sauté for 5 minutes

Add the crushed garlic

Sauté 5 more minutes

Meanwhile, in a medium sized bowl, whisk together the eggs, milk, flour, baking powder, oregano and 1/2 tsp salt until fully combined

Pour egg mixture into the skillet

Working carefully with a wooden spoon so you do not spill over the sides, mix egg mixture with caramelized onions and eggplant

Add the cheese, fold into the egg mixture

Carefully move the skillet from the stovetop to the middle rack of the oven (use an oven mitt!)

Bake for 10-12 minutes until the egg is set

Cool for at least 5 minutes before slicing

Serve topped with arugula, a sprinkle of salt and parmesan cheese (optional)