Preheat the oven to 375 degrees F. Using a melon baller, carefully scoop out a little spot in the center of each zucchini round for the filling to sit, making sure to leave about a 1/2-inch "floor" at the bottom. Put the scooped zucchini centers on a cutting board, season with salt and pepper and chop.
Heat the oil in a large skillet over medium heat. Add the scallions and cook, stirring occasionally, until softened, about 2 minutes. Add the garlic and cook, stirring frequently, until fragrant, 30 seconds to 1 minute. Add the mushrooms and chopped zucchini centers and cook, stirring frequently, until the mushrooms release their liquid and the zucchini start to soften, 5 to 7 minutes. Stir in the spinach and cook until wilted, 1 to 2 minutes. Remove from the heat and drain off any excess liquid. Stir in the ricotta and season with salt and pepper.
Stuff the zucchini cups with the filling and bake on a rimmed baking sheet until the zucchini is cooked through, 25 to 30 minutes.