Combine carrots, ¼ cup vinegar, 1 tablespoon water and ½ kosher salt; set aside.
Cut tofu horizontally into 2 planks, wrap tofu in a clean absorbent kitchen towel and set something heavy on top (like a cast iron skillet), let tofu press 15 minutes.
Heat 2 tablespoons oil in a large cast-iron or nonstick skillet over medium-high until shimmering.
Grate onion on a large-hole box grater then, using your hands, gently squeeze out excess liquid. Onion should measure about ½ cup. Add onion to sillet with oil and cook, stirring occasionally until golden brown, about 5 minutes.
Crumble tofu into ½- to 1-inch pieces.
Add tofu and remaining tablespoon oil, season with salt then toss to coat. Cook without moving too frequently (this is how we’ll get it crispy), until tofu starts to brown and crisp, about 5 minutes.
Add ginger and continue to cook until tofu is very crisp and browned, about 5 minutes. Add garlic and cook 1 minute.
Whisk together honey, gochujang, miso, sesame oil, and remaining 1 tablespoon vinegar. Add to skillet and stir to columbine.
Off heat stir in cucumber; season with salt to taste.
Serve in lettuce cups with quick-pickled shredded carrots.