Cook Vegetables and Chicken:
Heat the olive oil in a large pot over medium heat. Add the onion, carrots, and celery, and sauté until the vegetables are softened, about 5 minutes.
Add the garlic and chicken pieces. Season with salt, pepper, and dried thyme. Cook until the chicken is no longer pink, about 5-7 minutes.
Thicken with Flour:
Sprinkle the flour over the chicken and vegetables, stirring to coat everything evenly. Cook for about 2 minutes to remove the raw flour taste.
Add Liquids and Pasta:
Gradually add the chicken broth and heavy cream, stirring constantly to create a smooth mixture.
Once the mixture comes to a simmer, add the pasta. Reduce the heat to low, cover, and let simmer until the pasta is cooked, about 10-12 minutes, stirring occasionally to prevent sticking.
Finish with Peas and Cheese:
Stir in the frozen peas and cook for another 2-3 minutes, until the peas are heated through.
Remove from heat and stir in the chopped parsley and grated Parmesan cheese.
Serve:
Taste and adjust seasoning if necessary. Serve hot, with extra Parmesan cheese sprinkled on top if desired.
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