Place the chopped broccoli florets, shredded carrots, diced yellow onion, and minced garlic into the bottom of a 4-quart or larger slow cooker.
Pour the chicken broth over the vegetables. Sprinkle evenly with kosher salt, black pepper, and smoked paprika. Stir gently to combine.
Cover the slow cooker and cook on LOW for 4-6 hours or HIGH for 2-3 hours, until the broccoli is completely fork-tender.
About 30-45 minutes before serving, remove the lid and add the softened cream cheese cubes and heavy whipping cream. Cover and let heat for 15 minutes.
Remove the lid and whisk the soup vigorously until the melted cream cheese is fully incorporated and smooth. If you prefer a thicker soup, stir in the cornstarch and water slurry now and let cook for 10 more minutes.
Turn the slow cooker heat OFF (or to Warm). Gradually stir in 2 cups of the freshly grated sharp cheddar cheese until completely melted and smooth.
Serve hot, garnished with the remaining 1/2 cup of shredded cheddar cheese.