To prepare sauce, combine first 3 ingredients in a small bowl, stirring until honey dissolves. Stir in peanuts.
To prepare rolls, heat oil in a large nonstick skillet over medium-high heat. Add ginger, garlic, and chicken; sauté 5 minutes. Combine chicken mixture, hoisin sauce, and chile sauce; chill.
Add hot water to a large, shallow dish to a depth of 1 inch. Place 1 rice paper sheet in dish. Let stand 30 seconds or just until soft. Place sheet on a flat surface. Arrange 1/4 cup lettuce over half of sheet, leaving a 1-inch border. Top lettuce with 1/4 cup chicken mixture, 3 cucumber strips, 3 bell pepper strips, and 3 mint leaves. Folding sides of sheet over filling and starting with filled side, roll up jelly-roll fashion. Gently press seam to seal. Place roll, seam side down, on a serving platter (cover to keep from drying).
Repeat procedure with remaining sheets, lettuce, chicken mixture, cucumber strips, bell pepper strips, and mint leaves. Slice each roll in half diagonally. Serve rolls with sauce.