While the potatoes are simmering, prepare the filling by heating oil in a 12-inch sauté pan over medium-high heat until it shimmers.
Sauté the onion and carrots in the heated oil until they start to caramelize, about 5 minutes.
Stir in the garlic and cook briefly until fragrant.
Add the lamb or beef, salt, and pepper, cooking until the meat is browned and fully cooked, which should take around 4-5 minutes.
Sprinkle the meat with flour and stir well to coat, cooking for an additional minute.
Incorporate the tomato paste, chicken broth, Worcestershire sauce, rosemary, and thyme, stirring to combine.
Bring the mixture to a boil, then reduce the heat to low, cover, and let it simmer gently for 10 to 12 minutes or until the sauce thickens slightly.