Preheat the oven to 425 degrees. Make the spring rolls: Cook the ham in 2 teaspoons vegetable oil in a small nonstick skillet over medium heat until crisp, 2 to 4 minutes. Add the ginger, garlic and scallions; cook until fragrant, about 1 more minute.
Toss the cabbage and carrot with the hot ham mixture in a large bowl. Add the crab, cilantro, chili-garlic paste, 1 teaspoon sesame oil, the soy sauce and vinegar; toss.
Wrap the spring rolls (see right). Pierce each roll with a skewer in a few places to prevent bursting.
Place a rack in a baking dish and brush with vegetable oil. Mix the remaining 2 teaspoons vegetable oil and 1/2 teaspoon sesame oil; lightly brush on rolls. Place the rolls on the rack; bake until golden on top, about 15 minutes. Turn the rolls; bake until golden and crisp, 8 to 10 more minutes.
Meanwhile, mix all the sauce ingredients in a small bowl. Serve with the warm spring rolls.