Heat 2 tablespoons oil over medium-high heat in a large Dutch oven/soup pot. Season chicken with 1/2 teaspoon salt and 1/2 teaspoon pepper then sear until golden, about 2 minutes per side. Remove to a plate but leave drippings.
Melt 3 tablespoons butter with 1 tablespoon olive oil over medium low heat. Once melted, increase to medium-high heat and add onions, carrots and celery; sauté for 4 minutes scaping up the golden bits on the bottom of the pan. Add garlic and sauté for 30 seconds; sprinkle in flour then cook, stirring constantly for 2 minutes (it will be thick).
Add chicken back to the pot along with rice, bouillon, all seasonings, 3/4 teaspoon salt, bay leaf and chicken broth. Cover the soup to bring to a simmer, then displace the lid so it’s partially covering the pot, with about a one-inch opening. Simmer the soup for 12-15 minutes or until chicken is tender enough to shred.
Remove chicken to a cutting board and shred into bite-size pieces once cool enough to handle (don’t add back to soup yet). Continue to simmer the soup for an additional 10 minutes, partially covered.
After 10 minutes, whisk cornstarch with half and half (you can whisk with as little as 1/2 cup, just enough so it dissolves) and add to pot. Continue to simmer the soup uncovered for an additional 10-20 minutes or until the rice is almost al dente, stirring often.
Stir in shredded chicken (or rotisserie chicken if using), and cauliflower and continue to simmer for 5 minutes or until cauliflower is tender and rice is al dente. Add peas and warm through, 1-2 minutes. Stir in additional broth or half and half if desired for a less “chunky” soup. Taste and season with additional salt and pepper if desired (I like more salt).