I will start of by saying that while I have multiple food allergies (including milk), egg is not one of them.
On both the Dream and the Fantasy the chefs have made me a very nice chocolate cake thing (usually with some fruit or chocolate sauce as a topping) multiple times for dessert in the MDR, using Namaste chocolate cake mix (
http://www.namastefoods.com/products/cgi-bin/products.cgi?Function=show&Id=20), which is egg free. Although the instructions say to use eggs, an egg replacer may be a viable option and Namaste even has suggestions for that (
http://www.namastefoods.com/docs/EggReplacerSuggestions.pdf ). The chefs should have a good idea of what can/cant be done and it still turn out ok. I know one of the baked goods they made for me they used apple sauce as an egg relacement (I assume they were making a batch that was also safe for some other cruisers).
On both ships I had a safe-for-me pizza made in the QS pizza area. However, the crust they used for mine was an Udi's crust, and those contain eggs. If you talk with your dining staff and explain she would e.g. like pizza for lunch the next day, they may have suggestions on how to accomplish that, either in an MDR or at the QS. I recently saw a picture of a kiddo who had a safe pizza made for them in the MDR -- ordering the day in advance might make that doable as it gives them ample time to create safe ingredients and crust. You can also ask at the QS and they may or may not be able to do it -- it will depend largely on the ingredients they have on hand.
At the burgers QS area I was able to get allergy friendly chicken fingers and safe french fries, but I don't know if the allergy chicken fingers have egg or not (they are battered, so it is possible they do; again, ASKING is the only way to find out).
One thing I found helpful: I carry "chef cards" with me whereever I go. For one allergu you could fit it on a business card. Mine are printed on 4x6 index cards. When I eat out I give one to the waiter and they are free to share it with the chef. I have a bunch so if I don't get it back that is fine. it helps with communicating my food allergies, especially in noisy environments. On ship, I use them at the dining changes time, the first night in the MDR, and when I order from the QS on the pool deck (which is VERY VERY noisy). I have some on hand for the MDRs at breakfast and lunch but since the last 2 cruises we just preordered those meals I did not need them there. Consider making some. There are lots of sites online that have info you can use and some have free templates.
Pre-ordering dinners the night before makes things really easy. I also now pre-order breakfast and lunch. Yes, it means we are a little less spontaneous, but it means I can get the food I want. Like safe Mickey Waffles every breakfast (the one day I forgot near the end of the cruise our server reminded me and asked if I wanted them!

). For breakfast and lunch we could choose any of the dining rooms that was open for that meal, including Cabanas, in whichto eat, so we shifted around to get some variety, though usually we went with the dining room our servers would be in for the meal.
One thing that was also helpful is with the
DCL app I could look at ALL the menus for EVERY day. This meant that each night at dinner I had already looked at the next night's menu and knew what I wanted; so when it came time to pre-order all I was doing was doublechecking the menu and ordering (a quick process) instead of reading the menu, deciding, and ordering (a long process). Also, if there was something I really wanted from another MDRs menu (eg Butternut Squash soup), I would ask if it was possible to have it (it always was).
As for bread: On our last 3 cruises they have made us the most delicious sourdough rolls for dinner each night, and quickly learned we loved them an ate a lot so kept us in good supply. I was quite content to have this as my bread for the entire cruise, as was my sister.
For salad dressings I used oil and vinegar as the salad dressings all had something I was allergic to in them (milk or a fishy like anchovies). I am tempted to bring my own small container of safe salad dressing on board and keep it in the room fridge and just bring a small amount to dinner each night...
SW