Heat a large skillet over high heat. Add in 1 pound ground beef, and cook until browned. The beef will probably stick (that's a good thing, we want some of those yummy, charred meaty bits).
Add in ⅓ cup all-purpose flour, and stir to coat. Then remove the beef from the skillet and place in the basin of a slow cooker. Return the skillet to the heat, add in a splash of the chicken broth, and using a spoon, scrape off all the delicious bits. Remove from heat. Pour broth and bits into the slow cooker, along with the remaining chicken broth.
Add in 1 large onion, 2 large carrots, 3 stalks celery, 3 cloves garlic, 4 medium baking potatoes, 2 bay leaves, ¼ teaspoon salt, and ¼ teaspoon black pepper to the slow cooker. Stir, and cook on high for 4-6 hours, or low 6-8 hours.
With about a half hour left in the cooking time, remove the bay leaves and discard. Then use a potato masher to lightly mash up the potatoes (no need for it be smooth, you're just looking to thicken the soup). Add in 6 ounces shredded sharp cheddar cheese, ½ cup half and half cream, and ½ cup sour cream. Taste and adjust seasonings, if necessary. Continue cooking until warmed through, then serve.
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Notes
If you want to save on prep time, chop the veggies all at once using a food processor. You can also skip the browning of the ground beef (although, it does add a nice grilled, cheeseburger-y flavor). Just sprinkle the raw ground beef into the bottom of the slow cooker, add the flour, and toss to coat. Add in the chicken broth, then follow the recipe as written.