I use a poached egg pan, such as you describe with 4 little removable egg wells. Almost Fill pan ( so the egg wells just seat not float) with water and bring to boil, then set to simmer. Put tiny pat of butter or margarine in each well and allow to melt. Pick up each well in turn and encourage the melted butter to cover each inner surface. Once all wells to be used are greased, add the eggs, then the lid, and watch. I use a blunt knife gently touched to the white about half way between the edge of the yolk and the well ( is it almost firm?) to check if the eggs are almost cooked.
Medium eggs usually fit ok.