Heat olive oil in a large skillet over medium heat.
Add the chicken and garlic. Season with salt and black pepper, then cook for about 6 to 7 minutes, stirring occasionally, until the chicken is lightly golden and cooked through.
Remove the chicken and garlic using a slotted spoon and set aside on a plate.
In the same skillet, add the sliced onions and season lightly with salt. Cook over medium heat for 18 to 20 minutes, stirring occasionally, until soft and lightly golden.
Stir in the balsamic vinegar and barbecue sauce. Continue cooking for another 15 minutes until the onions are deeply caramelized and rich in color.
Meanwhile, cook the linguine according to package directions until al dente. Reserve 1 cup of pasta cooking water before draining.
Add the heavy cream, lemon juice, and parmesan cheese to the caramelized onions. Stir well and bring to a gentle simmer.
Add about ¼ cup reserved pasta water to loosen the sauce, then toss in the cooked linguine. Add more pasta water as needed to reach a silky consistency.
Remove from the heat and top with the cooked chicken. Finish with fresh parsley and extra parmesan before serving.