There's nothing inherently feminine about cooking; men have to eat too and it's really shameful when a man can't cook anything but ramen noodles, TV dinners, and microwave macaroni and cheese.
The issue is dividing the daily/weekly household duties into those suitable for man and the duties more suitable to women. Husband and wife are each given 24 hours every day, and those hours can only be spent once each.
Men should take care of the interfacing with the rest of the world (including supporting the family). Men also have the primary responsibility for pretty much everything. He has to make sure that everything is running smoothly. Women are more concerned with things domestic; their share of the duties should be in that sphere.
You want to divide up the duties in such a way that a natural man-woman relationship can exist. To try to illustrate, let me give you an example of the opposite of what you want: if the wife earns 100% of the income and the man stays home to take care of the children, that is going to be near-impossible. It's going to warp the natures of both.
But making a big deal about doing a load of dishes or even changing a diaper once in a while seems (to me) to be overcompensating for an insecurity somewhere. Kind of like how some guys aren't secure enough in their manhood to wear a pink shirt once in a while.
In a similar vein, men who really know how to fight -- those who are truly tough -- don't usually go around bragging about it. It's no big deal to them that they can kill with a single punch, deflect and subdue any attacker with their black-belt martial arts skills, etc. It's as natural as breathing for them, and it's such an obvious fact that they don't talk about it a lot.
It's the insecure gangster wimps that have to repeat in a litany over and over how tough and great they are (for example, rap music).
A real man will teach his KIDS to work and do dishes, and just supervise that they get done. And if his kids are gone/sick/etc. and his wife is engaged in some lawful activity, he will do the dishes just because they need to be done, and won't begin producing less testosterone as a result of the experience.