Raw Eggs in Hot Black Tea: Common in Ukraine? Correct Preparation?
I think what you've stumbled across here is an interesting implementation of a technique, and not really a specific dish. The technique (simplified) is poaching. It's just a very unique way of doing it.

I've had eggs poached in black tea in Macau, and you'll also find the technique used in English and French cooking. There's an old recipe for scallops with demi-glace that calls for the scallop to be garnished with a quail egg poached in tea with bergamot (Earl Grey, commonly). I wish I could remember the name of it and my French isn't nearly good enough to search effectively.

My eggs were cooked until just runny, but it's not uncommon to have them warmed just to hot and gooey with the white still translucent (and delicious with noodles).

There's probably a colloquial name for what she described, but it's essentially an adaptation of a technique that (very likely) originated in China.