Okay, I went back to reread OP's post.
It's possible that your DH prefers the nanny's cooking. While the timing and phrasing is terrible, voicing an opinion, and asking you to get her recipe says that he likes the way you cook, but prefers that recipe. Perhaps you could point out the timing and phrasing as issues to him, but get the recipe.
The rest of it is ridiculous. You both pay for the nanny, she's there to teach, love and nurture your children. Yes, there are times when very young children prefer the nanny, because she spends more time with them, but they outgrow it. Even then, there's is no reason for a spouse to act this way. It's definitely emotional abuse, and if it doesn't stop, perhaps you need to consider a separation or divorce.