Spread the tofu over a paper towel–lined baking sheet and let it drain for 20 minutes, then gently press dry with more paper towels. Season the tofu with salt and pepper. Heat 2 teaspoons of the oil in a 12-inch nonstick skillet over high heat until just smoking, then add the tofu and cook, turning occasionally, until browned on all sides, about 8 to 10 minutes; transfer to a bowl.

Heat the remaining tablespoon of oil in the now-empty skillet over medium-high heat until shimmering. Add the mushrooms and cook until browned, about 4 minutes. Stir in the chili-garlic sauce, garlic, and ginger and cook until fragrant, about 30 seconds, then pour in the broth.

Break the ramen into chunks and add to the skillet. Bring to a simmer and cook, tossing the noodles constantly with tongs to separate them, until they are just tender but some liquid still remains in the pan, about 2 minutes.

Stir in the vinegar, soy sauce, and sugar. Add the spinach one handful at a time, stirring until wilted and the sauce has thickened slightly. Return the tofu to the skillet and heat for 30 seconds until warmed through, then serve immediately.