Bring a small pot of water to a boil and cook the egg for 6.5 minutes, then immediately transfer to an ice bath.
Cook the soba noodles according to package directions, drain, and set aside.
Press the tofu to remove excess moisture, cut into cubes, and sear in a pan with half the sesame oil until golden.
In a medium pot, sauté the minced garlic, ginger, and sliced mushrooms in the remaining sesame oil until fragrant.
Stir in the vegetable broth, tamari, and gochujang, bringing the mixture to a gentle simmer.
Add the baby bok choy and edamame to the broth, cooking for 3 minutes until the greens are tender.
Divide the noodles and broth into a bowl, then top with the seared tofu, halved egg, scallions, and a squeeze of lime.