Preheat a grill pan or outdoor grill to medium-high heat. Pat the chicken thighs dry, rub with olive oil, 1/2 tsp salt, and pepper, then grill for 6 to 7 minutes per side until the internal temperature reaches 165°F (74°C). Let the chicken rest for 5 minutes, then slice.

Boil a large pot of water with the remaining 1/2 tsp salt. Cook the fettuccine until al dente (10 to 12 minutes). Before draining, reserve 1 cup of pasta water. Drain the pasta and set aside.

In the same pot over medium heat, melt the butter. Add the garlic and cook for 30 seconds. Pour in the heavy cream, bring to a gentle simmer, and cook for 2 to 3 minutes until slightly thickened. Remove from heat, whisk in the parmesan gradually, then stir in a few tablespoons of pasta water until the sauce is glossy and smooth.

Slice the rested chicken. Add the drained fettuccine and chicken to the sauce, toss with most of the parsley, and serve immediately with the remaining parsley on top.