In a frying pan, add your olive oil and mushrooms and sauté until fragrant. Add your asparagus to the pan and sauté and gently tossing over medium heat until almost fork tender. Then add your sliced leeks to the pan. Salt and pepper to taste and sauté until all vegetables and mushrooms are tender and vibrant green. Set aside.
In the meantime, bring a saucepan of water up to a slight boil/simmer 160 to 180 degrees F. (71 to 82 degrees C.) The best technique is to bring your water to a boil and then bring back down to a simmer. You want to use a pan/pot that is at least 4 inches deep and we also prefer a pot with a non-stick bottom so the egg does not stick to the bottom of the pan. If you are poaching 4 eggs then you need to choose a deeper frying pan/Dutch oven so that you can get all the eggs together without touching. Or you can poach them all separately, one at a time. If you are new to poaching, doing one egg at a time until you get the hang of it is optimal.
Add vinegar to the water. Vinegar has its pros and its cons. Eggs are made up of mostly protein and when the egg whites hit the hot water they tend to disperse in the water. If you add vinegar to the water, it helps the egg whites firm up faster so that your inner egg yolk has a chance to cook. However, vinegar, in some cases can make your poached egg have a firmer outside texture. So maybe try one both ways and do the way you are most comfortable with.
Break your egg into a small dish, carefully to avoid breaking the egg. This step just makes it easier to place the egg into the pan gently and if you get any shell in there it is easier to get out. Next stir your light boil/simmering water in a clockwise direction to make a whirl pool. This helps the egg whites wrap around each other to obtain that classic poached egg shape. Next, carefully, slide your egg white and whole egg into the prepared pan. Set your timer for 3 minutes. Ladle out your egg and if desired trim any unwanted edges on your egg. Season your eggs with salt and pepper as desired.
Place your sautéed asparagus, mushroom and leek mixture out onto a serving plate. Top with your poached eggs and sprinkle with a little freshly grated parmesan cheese if desired. Break open the delicious drippy egg and swirl around your delicious vegetables in the dreamy sauce and enjoy.