In a large saucepan or dutch oven, melt the butter over medium-high heat. Add the leeks, season with salt and cook until soft, about 5 minutes. Add the beef and season with salt and pepper. Cook, breaking up the meat with a wooden spoon, until no pink remains, about 8 minutes.
Add the wine and cook until mostly reduced, about 5 minutes. Stir in the chicken broth and lemon peel and bring to a simmer. Cover and cook over low heat until the meat is tender, 20 to 25 minutes.
Stir in the cream and return to a simmer over medium heat. Cook until the sauce is thickened, 10 to 12 minutes. Stir in 1/4 cup mint.
Divide the spaghetti, sauce and the remaining mint among 6 bowls.