Preheat oven to 375 degrees Fahrenheit.
Wash and prepare the greens: thoroughly wash the greens either by rinsing as necessary under cold water or filling a sink with cold water and submerging the leaves in water, tossing the leaves around to remove any sand or other debris which will fall to the bottom of the sink. Spin in a salad spinner or pat dry with towels to remove as much moisture as possible (some is ok, as long as the leaves aren't soaking wet).
Depending on your combination of greens, you can thinly slice the Swiss chard stems and leave any tender stems on the spinach, but remove the tough, center stems from the mustard greens, collard greens, and kale. Do this by either cutting down the length of the stem vertically or tearing the greens from the stem. Discard the stems.
Slice the larger greens into ribbons (chiffonade). An easy way to do this is to stack 3-4 mustard greens, collard greens, kale, or swiss chard leaves on top of one another and roll up from the long edge (like a cigar), then slice across to make long ribbons.
In a small bowl, combine 1/4 cup Parmesan, 1/4 cup mozzarella, 1/4 teaspoon each pepper and salt (or to taste), and crushed cracker crumbs.
In a large bowl, crack and beat eggs. Whisk ricotta cheese, remaining parmesan, and remaining mozzarella with the eggs. Add 1/2 teaspoon each pepper and salt (or to taste), whisk again and set aside.
Melt 2 tablespoons butter in a large skillet or saute pan. Add chopped onions and garlic, and cook until translucent, about 4 minutes. If using, add mushrooms and cook for another 4 minutes.
Add washed and shredded greens, one handful at a time, to the pan, stirring to combine. Cover the pan and let the greens wilt, approximately 5-6 minutes. Remove pan from burner and let cool about 5 minutes.
Add the greens to the bowl with the egg and cheese mixture, stir to combine. Transfer to your prepared baking dish, and top with the crushed crackers and cheese mixture.
Cover with foil and bake 30 minutes. Remove foil and bake an additional 15 minutes or until hot and bubbly.