Preheat the oven to 375 degrees F. Stir 2 tablespoons of the melted butter into the cracker crumbs. Press the crumb mixture into the bottom and 1-inch up the side of a 9-inch glass pie plate. Bake the crust in the center of the preheated oven until it begins to brown, about 7 minutes. Remove the crust from the oven and reduce the oven temperature to 350 degrees F.
Meanwhile, grate the zucchini, preferably using the grating attachment of a food processor. Toss the zucchini with 1/2 teaspoon salt and let drain in a colander for 10 minutes. Heat the oil in a medium skillet over high heat until hot. Reduce the heat to medium; add the onion and cook, stirring occasionally, until softened, about 5 minutes. Squeeze the zucchini well with your hands to remove excess moisture. Add the zucchini to the onion and cook over high heat for 3 minutes.
Stir together the flour, the baking powder, and 1/4 teaspoon salt in a small bowl until combined. In a large bowl with an electric mixer, beat the eggs until thick and fluffy, about 3 minutes. Add the Monterey jack and cottage cheeses, the flour mixture, and remaining melted butter to the eggs and beat well. Stir in the zucchini mixture and chiles and pour the mixture into the crust. Bake the quiche in the center of the oven until the top is puffed and golden brown and a cake tester inserted in the center comes out clean, 35 to 40 minutes. Set aside at room temperature 5 minutes, cut into wedges, and serve with salsa, if desired.
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