Sauté Aromatics: In a large skillet over medium heat, melt the butter. Add the diced onion, carrots, and celery. Sauté for 5-7 minutes until softened, stirring occasionally. Add the minced garlic and cook for another minute until fragrant. Transfer the sautéed vegetables to your slow cooker.
Combine in Slow Cooker: Add the chopped broccoli florets, vegetable broth, and milk to the slow cooker with the sautéed vegetables. Stir gently to combine all ingredients.
Slow Cook: Cover the slow cooker and cook on low for 3-4 hours or on high for 2 hours, or until the vegetables are very tender. The broccoli should be easily mashable.
Prepare for Creaminess: Once the vegetables are tender, use an immersion blender to partially blend the soup to your desired consistency. You can also carefully transfer a portion of the soup to a regular blender and blend until smooth, then return it to the slow cooker. Leave some chunks for texture if preferred.
Add Dairy and Seasoning: Stir in the heavy cream, freshly grated sharp cheddar cheese, salt, black pepper, and optional nutmeg. Stir continuously until the cheese is completely melted and the soup is smooth and creamy. Ensure the soup is not boiling when adding the cheese to prevent clumping.
Thicken (If Desired): If you prefer a thicker soup, stir in the cornstarch slurry (2 tablespoons cornstarch mixed with 2 tablespoons cold water) and cook on high for an additional 15-20 minutes, stirring occasionally, until the soup reaches your desired thickness.
Serve Warm: Taste and adjust seasonings as needed. Serve your comforting Broccoli Cheddar Soup hot, garnished with extra cheddar cheese, croutons, or fresh herbs.