SEASON CHICKEN: Remove broccoli from the freezer to begin thawing so it's easy to chop. Cut chicken breasts in half widthwise to make four thinner filets. Use a meat mallet/frying pan/your fist to give each piece a quick pound (Don't flatten chicken; we just want all the pieces fairly even in width). Season both sides. (I add 1/4 teaspoon of salt & pepper per side and 1/2 teaspoon garlic powder and Italian seasoning per side.)
COOK CHICKEN: Add 2 tbsp olive oil to a large, wide, heavy-bottomed pot. As soon as the oil is hot, add the seasoned chicken pieces in one layer. Cook for about 3-4 minutes per side or until chicken is cooked through (registers 160 degrees/juices run clear). Remove to a plate, tent with foil, and rest for about 5 minutes before thinly slicing, dicing, or shredding with 2 forks.
VEGGIES: Meanwhile, back to the pot (don't rinse it out!): reduce heat to medium and add in remaining 1 tbsp oil plus 1 tbsp butter. Once butter is melted, add in onion, celery, carrot. Season to taste (I add 3/4 tsp salt & 1/2 tsp pepper). Sauté, stirring frequently, scraping up the browned bits left by the chicken, until very tender, 6-8 minutes. (These veggies won't get much more tender so take time here.)
SEASONINGS: Add in minced garlic, chicken bouillon, and remaining 1 tsp Italian seasoning. Stir for 30 seconds.
THICKEN: Add remaining 4 tablespoons butter. Stir until melted. Sprinkle 1/3 cup flour evenly over everything. Cook, stirring constantly, for 1 minute. Gradually add in the 3 cups chicken stock, stirring until smooth, taking time to really scrape the bottom of the pot. Add in 3 cups milk. Increase heat to high and bring to a boil. Stir frequently (and continue to scrape the bottom well) to keep the base from sticking. Once boiling, reduce the heat to a simmer. Add in frozen broccoli, frozen peas, and frozen corn. Cook 2-4 minutes longer or until veggies are tender and warmed through.
FINISH: Once veggies are tender, turn off the heat. Add in the cheese, a handful at a time, and gently stir through until melted. Mix in the heavy cream. Finally, add the chicken (along with any juices left behind). Stir through just to rewarm. Taste soup and adjust seasoning to personal preference.
SERVE: Serve warm with freshly chopped parsley or thyme and crusty warm bread!