Bring garlic, ginger, and 4 cups water to a boil in a small saucepan. Reduce heat and simmer for 10 minutes before whisking in the soy sauce and wakame. Set broth aside.
In a large skillet, heat oil over medium-high heat. Add mushrooms and cook, tossing occasionally, until golden brown and crisp, 10–12 minutes. Season with salt and pepperTransfer to a large plate.
Meanwhile, cook soba according to package instructions in a large pot of boiling salted water. Stirring occasionally and remove when al dente. Drain and return reserved broth to a simmer.
Divide soba and broth among bowls; top with mushrooms, turnips, and egg yolks. Add scallions and sprinkle with sesame seeds.