In a slow cooker, add onion, broccoli florets, carrots, chicken broth, salt, pepper, paprika, and thyme. Stir to combine. Add chicken breasts in a single layer on top.
Cover and cook on low for 4 to 5 hours, or until chicken is cooked through. Cooking time will depend on thickness of chicken beasts.
Remove chicken breasts, shred with a fork, then return chicken to slow cooker.
In a bowl, combine cornstarch with about ¼ cup of your half & half and whisk into a smooth slurry. Add cornstarch mixture, remaining half & half, and shredded cheese to slow cooker and stir to combine.
Cover and cook for an additional 30 minutes on low before serving.
I recommend using freshly shredded cheese from a block, not pre-shredded bagged cheese. Pre-shredded cheeses are coated in anti-clumping powders that will make your soup grainy when melted.
Half & half can be substituted with whole milk for a slightly thinner soup.
Leftovers will keep for 2-3 days in a sealed container in the refrigerator.
I do not recommend freezing dairy-based soups, they tend to separate when thawed.