Step 1

Stir ¼ cup soy sauce and brown sugar in a medium bowl. Cut tofu into ½"-thick slabs and then into squares. Pat dry with paper towels. Add tofu and toss to coat. Let sit, tossing occasionally, 20 minutes.

Step 2

Meanwhile, remove green tops from scallions and very thinly slice; set aside. Coarsely chop white parts and place in a medium pot. Add mushrooms, kombu, ginger, garlic, and 8 cups water and bring to a simmer over medium-high. Reduce heat to low and cook 10 minutes. Remove from heat; discard kombu. Let dashi sit 20 minutes (this will coax more flavor from aromatics), then fish out and discard scallions, mushrooms, ginger, and garlic.

Step 3

Heat vegetable oil in a large skillet, preferably nonstick, over medium-high. Cook tofu, working in batches if needed, until golden brown and crisp around edges, about 3 minutes per side. Transfer to paper towels to drain; season with salt.

Step 4

While tofu is cooking, bring dashi to a boil. Add bok choy and carrots and cook until tender, about 3 minutes. Add remaining 2 Tbsp. soy sauce; taste and season with more salt if needed.

Step 5

Ladle soup into bowls and add tofu, radishes, and reserved scallion tops; drizzle with sesame oil.

Step 6

Do Ahead: Dashi can be made 3 days ahead. Let cool; cover and chill, or freeze up to 3 months.