Heat oven to 375ºF.
Remove any large stems from the greens and wash them thoroughly in a sink filled with at least 5 inches of water. Move the leaves around in the water, then let them sit for a few minutes to allow any sand or dirt to fall to the bottom of the sink. Once clean, roughly chop the greens. You should have 1 pound of greens once they are stemmed. (Weigh the greens after stemming, but before washing.) After washing the greens, place them in a salad spinner to thoroughly dry them.
Butter a 9-by-11-inch or 2 1/2-quart baking dish and set aside.
In a large mixing bowl, whisk together the eggs, ricotta, Parmesan, salt, and pepper. Set aside.
In a large, 13-by-11-inch roasting pan set over 2 burners on medium heat, melt the butter in one corner of the pan. Add the garlic, mushrooms, and a pinch of salt and cook until the mushrooms give up their liquid, 5 to 6 minutes. Add the greens and cook until they are wilted, 3 to 4 minutes. The greens will reduce to less than 1/4 of their original volume and begin to look like thawed, frozen spinach. Remove the pan from the heat.
Add the greens to the egg and cheese mixture and stir to thoroughly combine. Pour into the prepared baking dish, top with the crackers, place on the middle rack of the oven, and bake for 35 to 40 minutes. Allow to cool for 5 minutes and serve.