Warm a cast iron skillet over medium-high heat and add the vegetable oil, coating the pan evenly.
Season both sides of the chicken breasts generously with salt and pepper.
When the skillet is hot, place the chicken breasts in and sear each side until golden brown, about 2 to 3 minutes per side, then remove from the heat.
In a bowl, whisk together the softened cream cheese and butter until the mixture is smooth and creamy.
Incorporate the cream of chicken soup, minced garlic, chicken broth, and Italian salad dressing mix into the cream cheese mixture; whisk until fully blended.
Lightly spray the crock pot with non-stick cooking spray and arrange the seared chicken breasts on the bottom.
Pour the prepared creamy sauce evenly over the chicken in the crock pot.
Cover and cook on low for 3 to 4 hours until the chicken reaches an internal temperature of 165°F and becomes tender.
Just before serving, prepare the angel hair pasta according to package directions until al dente, then drain.
Serve the chicken whole or shredded atop the pasta, spooning the sauce over. Optionally, garnish with fresh parsley, cracked pepper, or grated Parmesan cheese.