Heat butter and olive oil in a large pot over medium heat. Add the onion and cook, stirring occasionally, until soft and golden, about 12-18 minutes. Add wine, bring to a boil, and reduce the liquid by half, about 8-10 minutes.
Add the pasta, 2 C of stock, and cook, stirring occasionally so the pasta does not stick to the bottom of the pot. Add the remaining stock a cupful at a time, as the pasta absorbs the liquid while you stir occasionally. Cook until the pasta is al dente, about 25-30 minutes. Remove from the heat. Add in the pecorino and stir to combine. Season with sea salt and pepper to taste. If it is too thick, add more stock until slightly runny.
In a large mixing bowl, toss your greens with olive oil, a little lemon juice, sea salt, and some black pepper. Divide the pasta among the bowls, top with the greens and more shaved pecorino. Finish with a drizzle of olive oil and some freshly ground black pepper.