Bring a large pot of salted water to a boil and cook the fettuccine according to package instructions; drain and keep warm.
While the pasta starts, heat 2 tbsp vegetable oil in a large frying pan over medium heat until shimmering, about 1–2 minutes.
Add 1 tbsp minced garlic and 1 tsp dried thyme to the pan and cook, stirring, until fragrant, about 1 minute.
Add the chicken breast pieces to the pan, season with salt and pepper to taste, and cook, stirring frequently, until the chicken is cooked through and no longer pink, about 5–8 minutes depending on piece size; transfer cooked chicken to a plate and set aside.
In a medium saucepan over medium heat, combine the 8 oz cream cheese, 1 stick butter, 1 cup Parmesan, and 1 cup whole milk. Stir continuously until the cheeses and butter melt and the sauce begins to bubble, about 3–5 minutes.
Remove the sauce from the heat and stir the cooked chicken into the sauce until evenly coated.
Plate portions of the cooked fettuccine and spoon the chicken Alfredo sauce over each serving. Serve immediately.