Prepare the grill baste, grill seasoning and lemon butter sauce. Once the sauce is done and sitting on low, prepare and start grill for cooking.
Grill Baste: Melt butter over medium heat in a medium saucepan. Add onion and cook until softened, about 2-3 minutes. Then add garlic, cook another 2-3 minutes. Sprinkle flour over onions and garlic and stir to combine and cook for 1-2 minutes, until browned. Stir in and whisk oil, vinegar, lemon juice, mustard, and sugar. Bring the baste to a boil then reduce heat to medium/low. Add parsley and cook for 5 minutes. Remove from heat and set aside. Be sure to whisk well before you use, as the baste will start to separate.
Cut each chicken breast into halves, making 8 chicken breast pieces. Season chicken with grill seasoning.
Then right before you are going to lay the chicken on the grill, brush the chicken grill baste. Add the chicken to the heated grill, close lid and cook for 5 minutes. Flip the chicken and cook for another 5 minutes, or until chicken is nicely browned, firm to the touch and juices are running. Cook chicken 10-15 minutes total until cooked thoroughly. I check my meat with a meat thermometer.
In the last minute of the chicken cooking top with one of the goat cheese rounds on each piece of chicken. Close the lid and let it cook for a minute. Then remove chicken from grill and set on a platter.
Stir Lemon Butter Sauce throughout chicken cooking.
Place 2 pieces of chicken breast on a plate, there should be 4 plates. Spoon sauce over each serving, to each their own, I say the more sauce the better. Sprinkle with basil and serve immediately.