Preheat the oven to 375 degrees F. Spray the cavities of a 12-cup muffin tin with the cooking spray.
Heat the olive oil in a large skillet over medium-high heat. Cook the onion, stirring occasionally, until softened, 4 to 5 minutes. Add the garlic and cook for 1 minute. Add the chard and mushrooms and cook until the chard wilts and mushrooms release their moisture, 5 to 6 minutes. Season with salt and pepper.
Whisk the eggs with 1/4 cup grated Parmesan cheese and a few dashes of hot sauce in a large bowl, preferably with a spout. Season with salt and pepper.
Evenly divide the chard mixture between the muffin cups. Pour the egg mixture over the top and stir each cup with a fork to make sure the ingredients are evenly distributed. Sprinkle with the remaining 1 to 2 tablespoons of grated Parmesan and bake until slightly golden and a toothpick or butter knife inserted in the center comes out clean, about 20 minutes.
Loosen the mini crustless quiches from the muffin cups with a rubber spatula and transfer to a platter for serving.
The mini crustless quiches can also be made-ahead, refrigerated for up to 3 days or frozen.