Heat olive oil in a large pan over medium-high heat. Add meat and cook until no longer pink, breaking it up with a wooden spoon and stirring frequently, about 5 minutes.
Add diced celery, onions, carrots, rosemary, thyme, salt, and pepper to the pan. Cook until the vegetables soften, about 5 minutes, stirring occasionally.
Add beef broth and tomato paste to the pan, stirring until the paste dissolves. Reduce heat slightly and let it continue to simmer until the sauce thickens, about 15 minutes, before turning off the heat. While it’s simmering, start preparing the mashed cauliflower in the next step.