Cook the pasta according to the package directions. Drain, toss in a little butter or oil and keep warm
Make the Carrot Beurre Monte Sauce
Cut the butter into tablespoon size pieces and have ready next to the stove.
In a saucepan over medium heat bring the carrot juice and cayenne pepper to a boil and cook, stirring constantly, until it is reduced by half.
Remove from the heat and add the butter, one piece at a time, whisking constantly. Add another piece of butter as the previous piece is almost fully incorporated. Keep whisking vigorously and adding butter until the sauce is thickened and glistens. Add salt to taste as needed.
Make the Filling
Heat the butter in a large skillet over medium heat until it foams up. Add the minced leek and cook until soft. Add the seafood and cook until done (or reheated if precooked). Add the dill and salt to taste.
Serve the pasta and seafood with the sauce and pass any extra sauce on the side.