In Malaysia, a common breakfast is roti telur + teh tarik which is close to the dark breakfast region. It's like paratha, with an egg, and milk tea.
It is difficult to put milk into food. Why not just drink it? Alternatively, can we drink eggs and flour?
Cheese is another variation for milk. What about grilled cheese and eggs? Or some variation on Mac and Cheese?
You can also consider other dimensions like vegetables and spices. According to this plane, shakshuka is pure egg. Add spices to milk and you have chai. Add eggs to chai and you have cursed eggnog.
Egg paratha is a common Indian dish. The dough can be made with flour and water, but is traditionally made with flour and milk.
https://www.thedeliciouscrescent.com/omelette-stuffed-parath...
We can certainly drink eggs, Rocky had a few in the morning before he went to punch hanging frozen meat.
There is prior art in the grilled cheese and eggs area: Denny’s Moons Over My Hammy.