Magda and Silly Pairs

Chef and his friend Magda have $2N$ mutual friends: $N$ of these friends are chefs and the other $N$ are chefettes. The chefs are numbered $1$ through $N$ and the chefettes are (independently) also numbered $1$ through $N$. Since Magda wants their friends to be as happy as possible and to preserve traditional family values, she wants to pair them up in such a way that each chef is paired with exac