In a bowl or large zip-top bag, combine the first six ingredients: 3 Tbsp lemon juice (juice of 1 lemon), 3 Tbsp olive oil, 1 Tbsp minced garlic, 1 tsp salt, 1/4 tsp black pepper, and 1/2 tsp dried basil or oregano. Mix well to make the marinade.
Add the 2 boneless skinless chicken breasts to the bag or bowl, coat them evenly with the marinade, seal or cover, and refrigerate for a few hours or up to overnight.
When ready to cook, preheat your grill to medium-high. Remove the chicken from the marinade and discard the used marinade.
Place the chicken on the preheated grill and cook until done (internal temperature 165°F / 74°C or no pink in the center). When finished, remove the chicken and let it rest for a few minutes before slicing.
While the chicken is grilling (or immediately after placing it on the grill), bring a large pot of water to a boil and cook 2 cups dried penne pasta according to package directions (about 10–12 minutes) until al dente. Before draining, scoop out and reserve 1/2 cup of the pasta cooking water, then drain the pasta.
In the same saucepan you used for the pasta, melt 2 Tbsp butter over medium heat. Whisk in 1 Tbsp minced garlic and the juice of 2 lemons (6 Tbsp) until fragrant.
Pour in 1/2 cup half-and-half (or heavy cream) and whisk until the mixture is hot and begins to steam (do not let it boil). Taste and add salt and pepper as needed.
Stir in 1/2 cup grated Parmesan cheese, 2 tsp dried basil, and 2 Tbsp capers. Heat gently, stirring, until the cheese is melted and the sauce is smooth.
If the sauce is too thick, add some of the reserved 1/2 cup pasta water a tablespoon at a time until you reach the desired consistency. Add the drained pasta to the sauce and toss to coat evenly.
Divide the sauced pasta among serving bowls, top with the sliced grilled chicken, and sprinkle additional Parmesan if desired. Serve immediately.