Heat the olive oil in a large Dutch oven over medium heat. Add the onion, carrots, and celery. Cook for 5 to 6 minutes, stirring occasionally, until the vegetables begin to soften.
Add the garlic and cook for 30 seconds, stirring constantly.
Sprinkle the flour over the vegetables. Stir and cook for 1 minute.
Slowly pour in the beef broth, stirring until the flour is mixed into the liquid.
Add the potatoes, thyme, rosemary, smoked paprika, salt, black pepper, and bay leaf. Stir to combine.
Increase the heat to medium-high and bring the soup to a gentle boil.
Reduce the heat to medium-low. Cover and simmer for 15 to 20 minutes, until the potatoes are tender when pierced with a fork.
Remove and discard the bay leaf.
Use a potato masher to gently mash some of the potatoes in the pot, leaving plenty of chunks.
Add the cooked prime rib and Worcestershire sauce. Stir and simmer over low heat for 5 minutes, until the beef is heated through. Do not boil.
Stir in the heavy cream and cook over low heat for 2 to 3 minutes, until warmed through.
Remove from heat and stir in the parsley.
Ladle into bowls and serve warm.
Notes
Tips and Tricks
Trim the Beef – Remove large pieces of fat from the leftover prime rib so the soup doesn't become overly greasy.Keep the Beef Tender – Since the prime rib is already cooked, avoid boiling it. Gentle heat keeps the pieces from becoming tough.Adjust the Thickness – Mash more potatoes for a thicker soup. Add a little extra beef broth if it becomes too thick.Watch the Salt – Prime rib may already be heavily seasoned. Taste the soup before adding any extra salt.Cut Even Potatoes – Keep the potato cubes about the same size so they cook evenly.
Storing and Reheating
Refrigerating – Store in an airtight container in the refrigerator for up to 3 to 4 days, counting from when the prime rib was originally cooked. Refrigerate promptly, within 2 hours of serving.Freezing – Freeze in airtight, freezer-safe containers for up to 2 to 3 months for best quality. The potatoes may become softer, and the cream may separate slightly after thawing.Reheating – Heat in a saucepan over medium-low heat, stirring occasionally, until the soup reaches 165°F. Add a splash of beef broth if needed.Thawing – Thaw frozen soup overnight in the refrigerator before reheating.