Sauté the vegetables. Melt 2 tablespoons of the butter in a large Dutch oven or heavy-bottomed soup pot over medium heat. Add the diced onion, carrots, and celery. Cook for 6-8 minutes, stirring occasionally, until the onions are translucent and the vegetables have started to soften.
Add the garlic and seasonings. Stir in the minced garlic, dried thyme, salt, and black pepper. Cook for approximately 30 seconds, stirring frequently, until the garlic is fragrant.
Add the remaining butter and flour. Add the remaining 2 tablespoons of butter to the vegetables and stir until completely melted. Sprinkle the flour evenly over the vegetables, mixing well to incorporate it into the butter. Continue cooking for 1-2 minutes, stirring constantly, until the flour is fully incorporated and no dry flour remains. The mixture will appear thick and slightly pasty.
Add the chicken broth. Slowly pour in the chicken broth, starting with a small amount and stirring thoroughly to loosen the flour mixture. Gradually add the remaining broth, stirring continuously until the broth is smooth. Increase the heat as needed to bring the soup to a gentle boil, then reduce to a simmer. Cook uncovered for 8-10 minutes, stirring occasionally, until the carrots and celery are tender and the broth has thickened slightly.
Stir in the half-and-half. Reduce the heat to low. Slowly pour in the room-temperature half-and-half, stirring until fully combined. The broth should become smooth and creamy while maintaining a relatively thin, soup-like consistency. Keep the heat low and avoid bringing the soup to a rolling boil.
Add the chicken and peas. Stir in the cooked, shredded rotisserie chicken and frozen peas. Continue cooking over low heat for 4-5 minutes, stirring occasionally, until the chicken and peas are heated through. Do not bring to a vigorous boil.
Season and serve. Taste the soup and adjust the salt and black pepper as needed. Remove from the heat, divide among six bowls, and garnish with fresh parsley, if desired. Serve immediately.