Melt 1 tablespoon butter with 1 tablespoon olive oil in a large skillet over medium-high. Add the onions with ½ teaspoon kosher salt and cook for about 25 minutes or until browned and caramelized, stirring frequently. Transfer the onions to a bowl and set aside.

Butterfly the chicken breasts, slicing in half horizontally to make 4 cutlets. Season the chicken with the remaining salt and ground black pepper. Add the remaining butter and oil to the skillet and reduce the heat to medium. Add the chicken to the pan and cook until golden, about 5 minutes. Flip the chicken and cook for another 3 minutes or until the chicken is cooked through. Add the chopped rosemary and garlic and cook, stirring, for 30 seconds. Remove the chicken from the pan and slice.

Meanwhile, in a large pot filled with 3 quarts of water and 1 tablespoon of kosher salt, cook the penne until just done, about 12 minutes. Reserve ¼ cup of the pasta water then drain the pasta. Add the pasta to the skillet with the sliced chicken and its juices, cooked onions, ¼ cup pasta water, and blue cheese. Season with the remaining kosher salt and black pepper. If the pasta seems dry, add more of the reserved pasta water.