Whisk together rice vinegar, ½ cup water, 1 tablespoon honey and ¾ teaspoon salt until salt has dissolved. Add cucumbers and set aside.
Heat 1 tablespoon oil in a large sauté pan over medium-high. Add carrots and ½ teaspoon salt; cook, stirring occasionally, until softened and starting to brown, about 10–15 minutes. Transfer to a bowl and set aside.
Heat remaining 2 teaspoons oil in now empty skillet over medium. Add onion, garlic and ginger; cook until starting to brown, 3 minutes.
Add beef and cook, breaking up into small pieces with a wooden spoon, until no longer pink; drain beef.
Return beef to sauté pan and stir in soy sauce, honey, gochujang, and mirin and cook until thickened, about 3 minutes; season with salt and additional gochujang to taste.
Serve in lettuce cups with pickled cucumbers, carrots, and rice.