In a large pot of boiling, salted water cook the pasta just until done. Reserve ¼ cup of the pasta water. Drain the pasta and toss with 2 tablespoons of the pasta water.
In a large nonstick skillet melt 2 tablespoons of the butter with 2 tablespoons of olive oil over moderately high heat. Add the onions and 1/2 teaspoon salt and cook, stirring frequently until well browned, about 25 minutes. Remove from skillet.
Add the remaining butter and olive oil to the skillet. When melted add the garlic and cook for 30 seconds. Add the caramelized onions, green onions, chicken, bleu cheese, remaining salt and pepper and pasta. Heat thoroughly. If the dish seems a little dry add a little of the reserved pasta water.
Variation: you may add fresh spinach or mushrooms after cooking garlic. Saute until both are wilted and proceed with directions of the recipe.