Melt the butter in a large, deep skillet. Add the chestnuts, season with salt and pepper and cook over moderately high heat, undisturbed, until browned on the bottom, 4 minutes. Transfer to a bowl.
Heat the olive oil in the skillet. Add the garlic and anchovy and cook over moderately low heat, stirring, until fragrant, about 4 minutes. Add the wine and cook over moderate heat for 1 minute. Stir in the leeks and carrot, add the stock and bring to a simmer. Cover and cook, stirring occasionally, until tender, about 15 minutes.
Stir the heavy cream into the leeks and season with salt and pepper. Fold in the chestnuts, transfer to a large bowl and serve.
Make Ahead: The recipe can be prepared ahead through Step Refrigerate the chestnuts and leeks separately.