Heat the butter: Set the Instant Pot to sauté mode and add 3 tbsp butter. Allow it to melt.
Add the vegetables: Once melted, add 1/4 cup chopped onion, 1 cup diced carrots, and 3/4 cup diced celery. Sauté until tender, about 5 minutes.
Incorporate the liquids: Stir in 2 cans (14 oz each) chicken broth, 2 large diced boneless skinless chicken breasts, 1 pkg (6 oz) Long Grain & Wild Rice, 1 tsp salt, 1/4 tsp black pepper, 1 1/4 tsp herbes de provence, and 1/4 tsp red pepper flakes. Mix well.
Pressure cook: Lock the lid of the Instant Pot. Set it to cook on high pressure for 5 minutes. Once the cooking time is up, let the pressure release naturally for 5 minutes, then do a quick release of any remaining pressure.
Thicken the soup: In a small bowl, mix 2 tbsp cornstarch with 2 tbsp cold water. While the soup is on sauté mode, slowly stir in the cornstarch mixture.
Finish with dairy: Add 4 oz cubed cream cheese, 1 to 1 1/2 cups milk, and 1 cup half-and-half to the pot. Stir gently to combine and heat through, being careful not to let it boil. Season to taste and remove the bay leaf.
Serve: Ladle the soup into bowls. For a delightful presentation, serve it in crusty bread bowls or alongside artisan bread!