Line a baking sheet with three layers of paper towels. Arrange tofu on paper towels then top with three more layers of paper towels. Set a second baking sheet on top and weigh down with something heavy, such as a Dutch oven or a few cans; let press for 15 minutes.
Meanwhile, for the pickled cucumbers, combine cucumbers, vinegar, ⅓ cup water, and 1 teaspoon kosher salt in a medium bowl. Let sit, scrunching with your hands every few minutes, until cucumbers are softened and lightly pickled, about 10 minutes or until ready to serve.
Meanwhile, bring a large pot of water to a boil. Season water with salt then add noodles and cook 1 minute less than specified by package instructions. Reserve ½ cup noodle cooking water; drain noodles and add back to pot.
Add onion, gochujang, hoisin, lime juice, toasted sesame oil, garlic, ginger and reserved ½ cup noodle water to a blender. Blend on high until completely smooth, about 1 minute.
Pour sauce into pot with noodles and toss gently to coat. Keep warm over low heat, stirring occasionally, until tofu is ready.
Whisk together cornstarch and ½ teaspoon kosher salt in a shallow dish. Cut tofu into ¼-inch thick strips. Dredge tofu in cornstarch to coat.
Heat ¼ cup oil in a 12-inch stainless steel skillet over medium-high heat until shimmering. Add tofu in an even layer and cook, without moving, until browned and crisp and naturally releases from skillet, about 5 minutes. Using tongs, flip pieces and continue cooking until golden brown, on bottom side, about 5 minutes more. Flip again and cook 5 more minutes until lightly golden browned all over.
Transfer tofu pieces to a paper-towel-lined plate and sprinkle lightly with kosher salt.
Adjust consistency of pasta with addition pasta cooking water as needed; season with salt and pepper to taste. Divide noodles and tofu strips between 4 bowls, top each serving with a handful of pickled cucumbers and fresh cilantro.