Heat the butter in a medium skillet over medium heat until melted. Add the onion and cook, stirring frequently, until it softens and begins to brown, about 4-5 minutes.
Add the garlic and cook, stirring, for 30 seconds. Remove from heat.
Place the chopped potatoes in the slow cooker. Transfer the cooked onion and garlic mixture, including any melted butter, into the slow cooker.
Add the thyme, salt, pepper, and broth. Stir to combine.
Cover and cook on high for 3-4 hours or low for 6-7 hours, until the potatoes are soft.
If using fresh thyme sprigs, remove and discard the stems (most of the leaves will have fallen into the soup).
Use an immersion blender to blend the soup to your preferred consistency, or transfer it in batches to a countertop blender. Leave some chunks of potato for texture, or blend until completely smooth.
Stir in the shredded cheddar cheese and the plain Greek yogurt or sour cream, if using.
Serve warm with your favorite toppings, such as crumbled bacon, chopped green onions, shredded cheddar, or sour cream.