In a large nonstick frying pan, melt the butter with 2 tablespoons of oil over moderately high heat. Add the onions and 1/2 teaspoon of the salt and cook, stirring frequently, until well browned, about 25 minutes. Remove.
Add the remaining 1 tablespoon oil to the pan and reduce the heat to moderate. Season the chicken with 1/4 teaspoon of the salt and 1/8 teaspoon of the pepper and add to the pan along with the rosemary. Cook the chicken until brown, about 5 minutes. Turn and cook until almost done, about 3 minutes longer. Add the garlic. Cook, stirring, for 30 seconds. Cover the pan, remove from the heat, and let steam for 5 minutes. Cut the chicken into 1/4-inch slices.
Meanwhile, in a large pot of boiling, salted water, cook the orecchiette until just done, about 15 minutes. Reserve about 1/4 cup of the pasta water. Drain the pasta and toss with 2 tablespoons of the pasta water, the onions, the chicken with pan juices, the blue cheese, and the remaining 1/4 teaspoon salt and 1/8 teaspoon pepper. If the pasta seems dry, add more of the reserved pasta water.