Brown the ground beef. In a large pot over medium-high heat, cook the ground beef until browned and fully cooked. Remove the beef from the pot and set aside.
Cook the vegetables. Add the diced onion, minced garlic, and frozen peas and carrots to the same pot. Cook until the onion begins to brown and turns translucent.
Add the flour and potatoes. Add the butter and allow it to melt. Stir in the flour, then return the ground beef to the pot along with the diced potatoes. Cook for approximately 2 minutes, stirring to evenly distribute the flour throughout the ingredients.
Simmer the soup. Pour in the chicken broth and add the fresh thyme sprigs. Bring the soup to a simmer while stirring, then reduce the heat to medium-low and cover. Maintain a gentle simmer for approximately 20 minutes, stirring every 5 minutes to prevent sticking.
Break down some of the potatoes. After about 10 minutes of simmering, the potatoes will begin to soften. As you stir, press some of the potatoes against the side of the pot to help them break down and thicken the broth. Leave some larger pieces intact for texture.
Add the cheese and cream. After the 20-minute simmer, remove the lid, remove the sprigs of thyme, and stir in the Velveeta cheese and heavy cream. Continue cooking until the cheese has melted, then cook for another 5 minutes to allow the soup to thicken again.
Season and serve. Taste and adjust the salt and black pepper as needed. Turn off the heat and allow the soup to cool slightly before serving.