Prepare the sauce mixture. Pour the half and half and chicken broth into a large measuring cup with a spout, then mix in the seasonings. Measure the remaining ingredients before you start cooking. Put a measuring cup inside the colander you’ll use for the pasta so you remember to save some cooking water later.
Cut and flatten the chicken. Slice the chicken breast in half lengthwise to create 2-4 thinner slices. Cover the pieces with plastic wrap and pound each side until about 1-inch thick, remembering that the chicken will plump up as it cooks.
Season the chicken. Pat each piece dry, then season both sides with salt/pepper and Italian seasoning. Sprinkle the ¼ cup flour over both sides and rub it across the surface to create the light coating used for searing.
Sear and rest. Heat 1-2 tablespoons olive oil in a large skillet over medium-high heat. Add the chicken and cook for 3-4 minutes per side, until a golden crust forms. Check that the chicken has reached a safe internal temperature of 165°F/74°C, then set it aside for 10 minutes before cutting it into strips.
Start the pasta water. Bring water to a boil for the pasta while you move on to the sauce.
Deglaze with the wine. Pour ½ cup dry white wine into the chicken skillet and lower the heat to medium. Scrape the bottom and sides with a silicone spatula to loosen the fond. Let the wine bubble gently for 3-4 minutes, until reduced by half.
Build the roux. Add 3 tbsp butter and 3 cloves garlic, minced, and cook for 1 minute. Stir in 3 tbsp flour and keep stirring continuously for 2 minutes so the raw flour smell cooks away.
Add the creamy liquid gradually. Pour the prepared chicken broth/half and half mixture into the skillet in small splashes, stirring continuously as you go. Work slowly so the roux remains intact and the sauce keeps its thick consistency.
Add the bouillon. Stir in the ½ chicken bouillon cube. Bring the sauce to a gentle bubble, then turn the heat to low and partially cover the skillet.
Cook the pasta. Salt the boiling water and add ½ lb. pasta, any kind. Cook only until no more than al dente, using a timer to avoid overcooking and stirring occasionally. Drain the pasta and reserve 1 cup pasta water.
Melt in the cheeses. Gradually sprinkle 1/3 cup Parmesan cheese, grated, and 1/3 cup Romano cheese, grated, into the sauce while stirring continuously.
Combine the pasta and chicken. Add the drained pasta to the skillet and gently toss it through the sauce with a silicone spatula. Add the sliced chicken along with any juices that have collected on its resting plate, then toss again to combine.
Adjust the consistency. If the sauce needs loosening, add the reserved pasta water in small splashes and toss gently until the consistency is where you want it.
Finish and serve. If desired, squeeze fresh lemon over the pasta just before serving. Garnish with parsley and serve.