While the tofu browns, slice the shiitakes. While the shiitakes brown, prep the garlic and ginger.
SAUTÉ TOFU: Pat tofu dry with paper towels and toss with 2 teaspoons of soy sauce. Heat 1 tablespoon of oil in 12-inch nonstick skillet over medium-high heat until just smoking. Lightly brown tofu on several sides, about 5 minutes total. Transfer tofu to clean bowl.
SAUTÉ MUSHROOMS AND AROMATICS: Add remaining tablespoon oil to skillet and return to medium-high heat until shimmering. Add mushrooms and cook until browned, about 4 minutes. Stir in chili sauce, garlic, and ginger and cook until fragrant, about 30 seconds.
SIMMER RAMEN: Stir in broth. Break bricks of ramen into small hunks and add to skillet. Bring to simmer and cook, tossing ramen constantly with tongs to separate, until ramen is just tender but there is still liquid in pan, about 2 minutes.
. THICKEN SAUCE AND FINISH: Stir in remaining 2 tablespoons soy sauce, vinegar, and sugar. Stir in spinach, one handful at a time, until spinach is wilted and sauce is thickened. Stir in tofu and serve.