1. Preheat the oven to 400 degrees F and set the racks in the middle and upper positions. Fill a wide saucepot with water 1 1/2 inches deep and bring to a gentle simmer over medium heat. Add in the white vinegar. Fill a large bowl with lots of ice and some water.
2. Mix the butter, miso, rice wine vinegar and ginger together in a large bowl. Season with salt and pepper and set aside.
3. Toss the mushrooms, asparagus, radishes and carrots with the olive oil and some salt and pepper. Spread them evenly over 2 baking sheets. Place them in the oven to roast until they are softened, 10 minutes. Then rotate the baking sheets' position and cook until just starting to slightly brown around the edges, another 10 minutes.
4. Crack the eggs into 4 separate dishes. Using a spoon, stir the simmering water to create an eddy effect. Gently drop the eggs into the center of the eddy, recreating it after each one if necessary. Let the water do the work and cook the eggs until the whites have set, 3 to 4 minutes each. Once the eggs are set, use a slotted spoon to remove the eggs and place into the ice bath to stop the cooking. Keep the poaching liquid warm and at the ready. This can be done up to 1 hour ahead.
5. When the vegetables are finished roasting, add them to the bowl with the miso butter and toss. Taste and adjust the seasoning.
6. Place a pile of frisee on 4 salad plates and top with the roasted vegetables. Drop the eggs back into the warm poaching liquid to take the chill off and place your perfectly poached egg on top of your veggies. Drizzle each egg with some toasted sesame if desired and sprinkle with some salt and pepper.
Cook's Note: If you can't find pencil asparagus for this dish, just use regular asparagus. Use the top 3 to 4 inches and quarter them lengthwise. The same goes for the baby carrots; just peel and slice regular carrots in half lengthwise and then in half crosswise on a diagonal.