1) In a large pot, add stock ingredients and bring to a slight simmer. Cover and let the stock simmer for about an hour. Strain out solids and return liquid to a simmer, uncovered, to let it reduce slightly.
2) In a medium skillet add a drizzle of olive oil over medium heat. Then add sliced mushrooms and cook until mushrooms lose their liquid and brown slightly. Keep warm.
3) Cook soba according to package in simmering water. Be careful not to overcook. When done, drain soba and toss with one tablespoon of sesame oil.
To make a bowl, divide noodles into bowl and ladle in hot mushroom broth to barely cover noodles. Season broth and noodles with a dash of soy sauce and rice wine vinegar. Top with sauteed mushrooms, grated carrots, sliced scallions, and soft boiled eggs. Drizzle bowl with more sesame oil.
1) Bring about 1/2 inch of water to a boil over medium high heat in a pot big enough to hold your eggs in one layer.
2) When water is boiling, add eggs straight from the fridge.
3) Cover pot and let cook for 6 1/2 minutes exactly.
4) Uncover pot and run cold water over eggs for 30 seconds.
5) Peel eggs starting with the larger end of the egg and slowly working up the egg. Cut eggs in half and serve in bowl.