Bring a large pot of salted water to a boil. Add the fettuccine and cook according to package instructions until al dente. Drain and set aside.

In a skillet, heat the olive oil and 2 tablespoons of butter over medium heat. Add the sliced onions and a pinch of salt. Cook, stirring occasionally, until the onions are golden brown and caramelized, about 20-25 minutes.

Add the minced garlic to the caramelized onions and cook for an additional 1-2 minutes until fragrant. Pour in the heavy cream and stir to combine.

Add the cooked pasta to the skillet and toss to coat in the sauce. Stir in the grated Parmesan cheese and remaining butter. Season with salt and pepper to taste.

Serve the pasta warm, garnished with chopped parsley and additional Parmesan cheese if desired.