For the sauce, in a small skillet over medium-high heat, combine stock, ginger and crushed garlic, and reduce by half. Remove ginger and garlic, and stir in peanut butter to melt and incorporate. Add Hoisin; remove from heat and stir in lime juice.
In a small bowl, season cucumber with salt and sugar. Toss and let stand.
In a large skillet over high heat, heat oil, 2 turns of the pan. When oil ripples and smokes, add chicken; brown and crumble. Add ginger, garlic, pepper, mild peppers and chili peppers; stir 2 minutes to cook peppers to tender-crisp. Add basil, mint, fish sauce, lime juice and scallions; taste and adjust flavors.
Serve in large platter topped with peanuts, with lettuce platter and sauce alongside.