Heat a grill on high. Keep one side on high and then set the other side to low heat. Scrape the grates clean if necessary.
Whisk together the olive oil, lemon juice, red wine vinegar, 1 1/2 tsp salt, sugar, garlic powder, oregano and 1 tsp red pepper flakes. Add the chicken breasts and the lemon (cut side down) to the marinade and allow to marinate for at least 20 minutes, turning the chicken a few times to coat.
Grill the chicken starting on the hot side of the grill for about 5 minutes. Flip and continue to cook on the low heat side of the grill until the internal temperature reaches 165 degrees F. Remove the chicken to a plate and tent with foil until ready to slice and serve.
Place the lemon halves on the hot side of the grill, cut side down, and grill for about 5 minutes. Remove the lemon halves from the grill and set on a plate, cut side up, until ready to use.
Cook the pasta according to package directions. While pasta is cooking heat a large sauce pan over medium heat and melt the butter. Add the clove of garlic and cook for about one minute until fragrant.
Squeeze the juice from the grilled lemons into the butter and garlic mixture.
Stir in the cream and cook until simmering. Stir about 3/4 cup of the Parmesan cheese, salt and red pepper flakes into the lemon cream. Stir until cheese is melted and turn heat to low.
Drain the pasta except for about 1/2 cup of the pasta cooking water. Stir the reserved pasta water into the Alfredo sauce and add the pasta to the pan, tossing to coat the pasta with the Alfredo sauce.
Transfer the grilled lemon Alfredo pasta to a large serving bowl and top with the remaining 1/4 cup of Parmesan cheese and chopped parsley.
Slice the grilled chicken and place on top of the pasta to serve.