In a large non-stick skillet over medium-high heat, melt ¼ cup of unsalted butter. Add the sliced onions and sauté for 25-30 minutes, stirring occasionally, until they are very tender and caramelized. Season with ¼ teaspoon kosher salt and ¼ teaspoon black pepper. Transfer the onions to a plate.

In the same skillet, melt ½ cup of unsalted butter over medium-high heat, stirring constantly. Once the butter turns chestnut brown and has a nutty aroma, remove from heat and let it cool slightly.

Bring a large pot of salted water to a boil. Cook the linguine until al dente, about 8-10 minutes. Reserve 1 cup of pasta water, then drain the rest.

In a food processor or blender, combine the caramelized onions, browned butter, and ½ cup of reserved pasta water. Blend until smooth.

In the original skillet over medium heat, add the blended onion mixture, heavy cream, and 1 cup of parmesan cheese. Cook until the cheese melts and the sauce is heated through. Add the cooked linguine to the skillet and toss to coat the noodles in the sauce. If the sauce is too thick, add more reserved pasta water to reach the desired consistency. Serve hot, garnished with additional parmesan cheese and chopped parsley.