Prick the potatoes all over with a fork and brush them with 1 tablespoon melted butter. Put on a microwave-safe plate and microwave, flipping halfway, until fork tender, about 20 minutes. (Alternately, bake in the oven at 400 degrees F on a baking sheet until cooked through and fork tender, about 1 hour.) Let cool slightly.
Cut a 1/4-inch-thick slice off the top of each potato. Carefully scoop out the flesh into a medium bowl and mash with the milk, sour cream, 2 tablespoons butter and 1 teaspoon salt using a fork or potato masher. Fold in the chives. Put the potato bowls on a baking sheet.
Preheat the oven to 400 degrees F. Meanwhile, melt 1 tablespoon butter in a large skillet over medium heat. Add the carrots and cook, stirring occasionally, until softened and lightly browned, 5 to 6 minutes. Add the beef and cook, breaking it up with a wooden spoon, until cooked through, about 4 minutes. Stir in the Worcestershire, tomato paste, thyme, 1/2 teaspoon salt and 1 cup water and bring to a boil, and then reduce heat and cook until the beef and carrots are coated in sauce with a little extra sitting on the bottom of the pan, about 1 minute. Stir in the peas until heated through, 1 to 2 minutes more.
Spoon the beef mixture into the potato bowls. Spoon or pipe the mashed potatoes on top of the beef mixture and bake until heated through and the potatoes brown on top, about 15 minutes. Cut the remaining tablespoon of butter into 4 slices and top each potato with one.
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