Step 1: Prepare your vegetables by dicing the potatoes and onion, mincing the garlic, and cutting broccoli into bite-sized florets if using fresh.
Step 2: Add diced potatoes, broccoli florets, diced onion, minced garlic, vegetable or chicken broth, dried thyme, salt, and black pepper to the basin of your slow cooker. Stir gently to combine.
Step 3: Cover the slow cooker and cook on the LOW setting for 6-8 hours, or on the HIGH setting for 3-4 hours, until the potatoes are completely fork-tender.
Step 4: Once the vegetables are tender, you can thicken the soup. For a chunkier soup, mash some of the potatoes directly in the pot with a fork or potato masher. For a smoother soup, carefully remove 1-2 cups of the soup mixture, blend it with an immersion blender or in a regular blender (exercise caution with hot liquids), and then return the blended portion to the slow cooker.
Step 5: Stir in the heavy cream and shredded sharp cheddar cheese into the slow cooker. Continue to stir until the cheese is completely melted and the soup is smooth and creamy.
Step 6: Taste the soup and adjust seasonings if necessary, adding more salt or pepper to your preference. Serve hot, garnished with extra shredded cheddar, fresh chives, or crispy bacon bits if desired.