Heat the olive oil in a large Dutch oven over medium heat.
Add the diced onion and diced carrot to the Dutch oven.
Sauté for 3 to 4 minutes, or until the vegetables begin to soften.
Add the minced garlic to the pot.
Season with Italian seasoning, red pepper flakes, salt, and black pepper.
Stir to combine and cook briefly until the garlic is fragrant.
Reduce the heat to low.
Add the butter to the pot.
Stir until the butter is fully melted and evenly combined with the vegetables and seasonings.
Sprinkle the flour over the mixture.
Stir continuously and cook for about 1 minute to remove the raw flour taste.
Slowly pour in the heavy cream while stirring constantly.
Continue stirring until the cream is fully incorporated and the mixture is smooth.
Slowly add the chicken broth while continuing to stir.
Increase the heat and bring the soup to a gentle simmer.
Simmer for 5 to 7 minutes, or until the soup begins to thicken.
Add the bowtie pasta and cooked shredded chicken to the soup.
Stir to combine.
Cook for 6 to 8 minutes, or until the pasta is fully cooked and tender.
Remove the Dutch oven from the heat.
Stir in the shredded mozzarella cheese and shredded Parmesan cheese.
Continue stirring until the cheeses are fully melted and the soup is creamy.
Taste and adjust seasoning if needed.
Garnish with fresh parsley before serving.
Serve warm.