Prepare the potatoes: Place cubed potatoes in a large pot of cold salted water. Bring to a boil and cook until fork-tender, about 10-12 minutes. Drain well and return to pot.
Make the mashed potatoes: Add butter and warm milk to potatoes and mash until smooth. Season with salt and pepper to taste. If using cheese, stir in half of it now. Set aside.
Preheat your oven to 375°F (190°C) and lightly grease a 12-cup muffin tin with cooking spray.
Prepare the filling: Heat olive oil in a large skillet over medium heat. Add onions and carrots, cooking until softened, about 5 minutes.
Brown the meat: Add ground beef to the skillet and cook until no longer pink, breaking it up with a spoon, about 5 minutes. Drain excess fat if necessary.
Add the aromatics and seasonings: Stir in garlic and cook until fragrant, about 30 seconds. Add tomato paste, Worcestershire sauce, thyme, salt, and pepper.
Create the sauce: Pour in beef broth, bring to a simmer, and cook until liquid is reduced and mixture is thickened, about 3-4 minutes. Stir in frozen peas and remove from heat.
Assemble the mini pies: Divide the meat mixture evenly among the muffin cups, filling each about halfway. Top with mashed potatoes, either spooning it on or piping it for a fancier presentation.
Add final touch: Sprinkle remaining cheese on top of each pie if using.
Bake until golden: Place in preheated oven and bake for 15 minutes, then broil for 1-2 minutes until tops are golden and slightly crispy.
Cool slightly before serving: Allow mini pies to cool for 5 minutes before carefully removing from the muffin tin using a knife to loosen the edges.