Heat oven to 375ºF. Lightly butter a 9-by-11-inch glass baking dish and set aside.
Melt the butter in a 12-inch sauté pan over medium heat. Add the mushrooms and garlic, and sauté 1 to 2 minutes. Add the beet greens and cook, stirring occasionally, until the greens begin to wilt, 2 to 3 minutes. Remove the pan from heat and season with salt and pepper, to taste. Set aside to cool slightly.
In a separate bowl, whisk together the eggs, ricotta, Parmesan cheese, and 1/2 teaspoon salt. Fold the cheese mixture into the slightly cooled greens. Pour into the prepared dish and top with crumbled crackers. Cover with aluminum foil and bake for 30 minutes. Uncover and bake until the crackers brown slightly, an additional 7 to 8 minutes. Remove from the oven and cool for 5 minutes before serving.