Preheat the oven; pre the chicken - Preheat the oven to 350℉ and spray your baking dish with olive oil. Shred the chicken and spread on the bottom of the baking dish.
Prep the leeks and mushrooms - Clean, destem, and slice the mushrooms. Trim and wash the leeks, being careful to rinse out any dirt between the leaves. Slice the leeks across. Add the sliced mushrooms to a dry pan. Cook until the mushrooms have expelled their water. Add the sliced leeks and 2 T of butter to the pan, and continue cooking until the leeks are very soft and the liquid is cooked off.
Make the batter - Add the eggs to a mixing bowl and whisk them. Stir in the flour, then the milk, a little at a time. You should have a pourable batter. Fold in the mushrooms and leeks. Pour the batter over the chicken, and poke it around using a fork.
Bake - Bake for 30-40 minutes, until the "pudding" is set and browned across the top.