Bring a large pot of salted water to a boil. Add the green beans and cook until tender and bright green, about 8–10 minutes. Remove with a slotted spoon and set aside.
In the same pot of boiling water, cook the orecchiette according to package instructions until al dente.
While the pasta cooks, steam the cauliflower florets until soft, about 10 minutes.
In a large skillet, heat the olive oil over medium heat. Add the diced onion and cook until soft and lightly golden.
Mash the cooked cauliflower with a fork or pulse it briefly in a food processor. Keep some texture.
Add the green beans and mashed cauliflower to the skillet with the onion. Stir to combine.
Drain the pasta, reserving about ½ cup of pasta water. Add the pasta to the skillet.
Toss everything together, adding pasta water as needed to create a light, silky sauce. Drizzle in more olive oil if needed.
Season with salt and pepper. Serve immediately with grated Parmigiano Reggiano.