1. 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.
2. 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.
3. 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.
Wine Recommendation: The onions and cheese drive the wine choice for this dish. A lighter red wine from the Piedmont region of Italy, such as one based on the barbera or dolcetto grapes, has the weight and acidity to stand up to the sweet and salty flavors.