Combine the chopped broccoli, carrots, onion, and garlic cloves in your slow cooker. Add the chicken stock, white wine, salt, pepper, and paprika. Stir to combine.
Set your slow cooker to Low and cook for 6 hours. Open lid and stir, testing the tenderness of your vegetables. There are two steps you can take here. If you prefer a creamier soup, grab your potato masher and mash the broccoli. If you prefer a chunkier version, you can simply leave as is.
Cover and cook on High for an additional 2 hours.
In the last 15 minutes of cooking, remove the lid of the slow cooker. Add the cheese to the soup in 1/4 cup batches, stirring a heckuva lot to ensure it doesn't clump at the bottom of your slow cooker. When one batch of cheese is fully melted into the soup, add the next. Repeat.
When all of the cheese is added and the soup is fully combined, stir in the half and half. Serve immediately from the slow cooker.