Add the oil to a large skillet over medium high heat and heat until shimmering. Add the beef, breaking it into large chunks and spreading it evenly across the skillet. Allow to brown, undisturbed, for several minutes, then stir, repeating on the other side. When all the beef is browned and no pink remains, drain off any excess fat. Season to taste with salt and pepper.
Add a splash of water or chicken broth to the skillet and use your spatula to scrape up any browned bits. Transfer the beef and all the juices to a slow cooker.
Add the garlic, carrots, potatoes, onion, celery, tomato paste, parsley, chicken broth, and milk to the slow cooker along with the beef. Stir well to combine.
Turn on high for 3.5-4 hours or low for 6-7 hours.
When the potatoes are tender, add the Velveeta cheese, put the lid back on, and allow to melt for about 20 minutes.
Add the sour cream and stir everything well. Taste and season with salt and pepper.
Serve with bacon, jalapenos, garlic bread or waffle fries.
Notes
Storage: Store in an airtight container in the refrigerator for up to 7 days. You can freeze this soup to serve for later. Place in an airtight, freezer safe container and freeze up to 3 months.