Add olive oil to medium skillet set over medium heat. Add diced potato and cook until tender on the inside and crispy on the outside, about 5 minutes, stirring frequently. Add in the onions and celery and cook until tender, about 3 more minutes.
Stir in the garlic and the shredded roast beef and cook until the beef is heated throughout.
Meanwhile, heat the roast beef drippings in a small saucepan set over medium heat until reduced by about half, about 8 minutes. Whisk in the cornstarch until well combined and then pour into the beef hash.