Salt and pepper the chicken breasts
Grill on medium heat until cooked through and grill marks on both sides
Meanwhile prepare the basil sauce
In deep skillet melt the butter and add the flour, whisking for 1-2 minutes
When bubbly, slowly whisk in the whole milk, about ½ cup at a time. Don't let it boil but let it get thick before adding the next ½ cup
Keep adding the milk until it is all incorporated. Cook on medium heat, whisking frequently, for another 5-10 minutes so it can thicken up. Do not let it boil
Whisk in the Parmesan cheese, pesto and salt and pepper. Let it cook for another 5 minutes or so.
Add the drained, al dente pasta to the pesto sauce and toss together.
When the chicken is done, cut into strips and put on top of the pasta
Serve with extra pesto if desired.