Washing cookware while mountaineering...

Repackage each Mountain House into a freezer ziploc bag at home. Write the cooking directions in sharpie on the ziploc. When at camp pour the boiling water into the dinner ziploc, then put your ziploc in a meal cozy for the 10+ minutes that it needs to sit. The meal cozy keeps it warm, and you can eat the dinner while keeping it in the cozy.

Meal cozys are extremely easy to make and weigh very little. I made mine without the fold lid in the picture.  You can also use your hat or put it in your puffy jacket if you don't want to make a cozy.

https://www.cloudlineapparel.com/blogs/cloudline/how-to-make-an-insulated-backpacking-meal-cozy

I did a 30 day trip in the winds and all we were fine just scrubbing our pots with snow or fingers then occasionally dipping everything in boiling water. Just make sure to keep your hands clean when working with food stuff and you should be fine!

Another system to consider:

Take a Mountainhouse for the first meal. After you're done eating it, save the bag. For every future meal (ramen, oatmeal, etc.), dump the food in the old Mountainhouse bag to rehydrate it. Those bags will easily last 10 days if you're careful. If you're not careful, or because the zippers do sometimes randomly fail, take 2 Mountainhouse bags and eat another one midway through for a 'fresh' bag.

I try to never eat food out of my water pots.

I also recommend making all your meals super watery. It helps keep you hydrated, it helps dilute the 200% sodium, and you aren't left with a container caked in food residue. Except for noodles, you can drink pretty much anything without using a spoon.

Hobo Greg wrote:

I remove the desiccant packet (weight savings) 

you...do...what??

do you also remove the small pebbles lodged in the soles of your shoes each day???  :)

I roll with a folding cup that I will use for oatmeal and stuff to keep my pot clean, it fits on my pot as a lid with my stove inside which is nice. 

If you're worried about bacteria build up,  just rinse everything with your daily ration of bourbon.

Spit on it, rub it down with your shirt... or scub your pots out with some water and a handfull of sand or snow, then rinse with water or snow.... or just lick it, or leave it alone and savor that tasty flavor of spicy ramen mixed with your morning oatmeal who gives a shit?

When did we (especialy the outdoors minded) get to the point where we have to waste sanitizer wipes and paper towels to clean a pan (PERIOD) but especially while backpacking?
Boil it out, or leave it alone... or carry a washcloth or the like as a dedicated kitchen wipe (REUSABLE), one for the entire trip - stop wasting disposables, killing trees and using chemicals for unnecessary reasons.

 Ill add  a +100 for clean up immediately, do it once while hot and wet and it takes no time or effort.

No real worry for bacteria in zubzero temps, or if you cook to a full boil, and not eat raw chicken.
 
 Also, who carries10 mountain houses for a 10 day trip?  Sure they might be convenient for a day or two, but Good Gravy Man, what a waste or plastic, bags, space and money.
Can one not buy bulk dehydrated soup from WinCO, or whole foods?  Eat "Pasta sides" and instant ramen with tuna, jerkey or protien powder (together or seperate)?  can you not buy bulk mtn house and transfer into smaller containers, re-usable tupperware or other bags?  also, for the money you spend on 10 mtn houses, you could just buy a food dehydrator, then make all your own stuff.... (google has all those recipes - although, maybe a forum for thats not a terrible idea).

someone said ramen is poor in nutrients, but powdered elecrolyte drinks, protien powder, chia seeds, flax seeds and the like added to oatmeat, ramen or water or any other meal/food isnt. neither is dried fruit or jerkey.  ramen is a good filler for hunger, nutrients can come from protein or granola bars, GORP or the above mentioned suppliments.