01 - Place broccoli florets, diced onion, carrots, celery, and minced garlic in the slow cooker. Pour vegetable broth over the vegetables, ensuring they are mostly submerged.
02 - Cover slow cooker and cook on low heat for 4 hours until vegetables are completely tender and easily pierced with a fork.
03 - Melt butter in a small saucepan over medium heat. Whisk in flour and cook for 1 to 2 minutes, stirring constantly until mixture bubbles and becomes fragrant but does not brown.
04 - Gradually pour half-and-half into the roux while whisking continuously. Continue whisking for 3 to 4 minutes until sauce thickens and becomes completely smooth without lumps.
05 - Pour the cream mixture into the slow cooker and stir to combine. Use an immersion blender to puree soup until desired consistency is reached, leaving some texture if preferred.
06 - Stir in shredded sharp cheddar cheese, salt, black pepper, smoked paprika, and ground nutmeg. Continue stirring until cheese is fully melted and incorporated throughout the soup.
07 - Taste soup and adjust seasoning as needed. Serve hot, garnished with additional shredded cheddar cheese or accompanied by crusty bread.