Unironic answer (for anyone browsing this who's curious) - misogyny would be if you decide that you need to distribute the chores. If the two of you come to an agreement together (and I mean in a spirit of partnership, not her just agreeing to whatever you propose) then you're in the clear. If you tell her she needs to do the cooking because she's a woman, that's misogyny. If you tell her she's not allowed to do the cooking because she's a woman, that's also misogyny.
Unironic answer (for anyone browsing this who's curious) - misogyny would be if you decide that you need to distribute the chores. If the two of you come to an agreement together (and I mean in a spirit of partnership, not her just agreeing to whatever you propose) then you're in the clear. If you tell her she needs to do the cooking because she's a woman, that's misogyny. If you tell her she's not allowed to do the cooking because she's a woman, that's also misogyny.