To a large pan on the stove over medium high heat, add the oil and tbsp of butter Once the butter has melted at the chicken and cook for 3 - 4 minutes on each side until cooked through. 

While the chicken is cooking bring a large pot of water to a boil, cook the pasta according to the package instructions in the boiling water. When the pasta is tender strain it and set it to the side. Once the chicken has cooked through remove it to a plate and set the plate to the side. Cut the chicken into smaller pieces.

To the same pan the chicken was cooked in add the remaining butter, minced garlic and tbsp of flour. Whisk until the flour no longer looks white.

Slowly whisk the 1/2 cup of chicken broth bring it up to a simmer, then slowly whisk in the remaining 3/4 cup of chicken broth and half-and-half. Bring this mixture up to a simmer and whisk well to combine the ingredients.

Once the sauce has thickened up stir in the cheese. Let the cheese melt down Stir in the cooked strained pasta and chicken.