Melt 1 tablespoon of the butter in a Dutch oven over medium-high heat. Add the ground lamb, 1/2 teaspoon salt and a couple turns of pepper and cook, breaking up the meat a little, until lightly browned, about 5 minutes. Transfer to a plate using a slotted spoon and reserve the pan drippings.
Add the remaining 3 tablespoons butter to the pot. Stir in the carrots, onion and thyme and cook until the vegetables begin to soften, about 5 minutes. Add the mushrooms and cook until softened, about 3 minutes. Sprinkle the flour over the vegetables and cook, stirring, until the flour is toasted, about 1 minute. Add the wine and simmer until almost completely absorbed, about 2 minutes.
Return the meat to the pan, add the broth and bring to a simmer. Cook until the sauce thickens and the meat is tender, about 15 minutes. Stir in the gnocchi and cook until warmed through, about 3 more minutes.
Ladle the stew into bowls and sprinkle with the mint.
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