Place the diced russet potatoes and chopped yellow onion into the bottom of your slow cooker. Add the frozen broccoli florets on top of the potatoes. Scatter the cubed Velveeta cheese and butter evenly over the vegetables.

In a large mixing bowl, combine cream of chicken soup, cream of mushroom soup, minced garlic, and chicken broth. Sprinkle in flour, salt, and pepper, and whisk everything together until the mixture is smooth.

Pour the soup mixture over the vegetables, cheese, and butter in the slow cooker. Gently stir everything together to ensure the soup base coats all the ingredients evenly.

Cover the slow cooker with the lid and set it to cook on LOW for 5-6 hours. Stir occasionally to help the cheese melt evenly and prevent any sticking to the sides of the slow cooker.

Once the cooking time is up, check the potatoes and broccoli to ensure they're tender. Taste the soup and adjust the seasoning with additional salt and pepper if needed.

Ladle the hot soup into bowls and serve immediately. For extra cheesiness, top each bowl with shredded cheddar cheese, a sprinkle of fresh parsley, or crispy bacon bits. Pair with a side of crusty bread or garlic toast.