In any event, all's well that ended well. We realized that it CAN be done - and here's how. Bring water to a boil in another pot or teakettle and pour it into the hot pan from a 450 degree oven. Make sure your dutch oven or large shallow pan has only about an inch of water in it. (We tried siphoning off water with spoons and cups. Not so great. Also, the egg whites were destroyed in the process.) Spoon in the whole eggs and cover. Place in the hot oven until the whites are cooked and the yolks are the way you like them. Not so bad.
This, is not how we did it. But they ended up fine anyway. We ended up taking so much time with the slow-poaching oven process that the hash was done. We moved the (non-stovetop) pan to the stovetop and slowly covered the eggs with boiling water.
For the hash, take a variety of loads of vegetables (old, new, wilted - whatever's in your frig): parsnips, beets, carrots, onion, shallot, scallions, lots of potatoes, celery, celery root, fennel. I'm sure you can think of more! Root veggies are great. We haven't used spinach or tomato yet. I'd use sweet potatoes although Tim doesn't. In any event, place them one by one in a food processor and either grate or process into small chunks about 1/4". You'll probably need to empty the food processor a few times. Then process ham or chorizo or whatever you'd like for a meat.
Place some olive oil and fresh garlic in two large frying pans or dutch ovens on the stovetop on high heat. Turn down heat and add in fresh and dried herbs and salt and pepper.We cook down a huge bowl (4 qts or more) of these veggies and meat in large saute or frying pans on the stovetop until it's all blended and cooked. Top with your poached or fried eggs. And serve with raisin bread toast!
