To make the marinade: Combine the scallions, ginger, lemon juice, lemon zest, oil, garlic, tamari, brown sugar, and ¼ cup water in a food processor or blender and puree until smooth, scraping down the sides of the bowl as needed.
Toss the mushrooms with half the marinade in a small glass bowl and toss the beans with the remaining half of the marinade in another small glass bowl. Cover and refrigerate for at least 2 hours or overnight.
Soak 12 wooden skewers in hot water for 10 to 15 minutes, then drain.
Drain the mushrooms and beans and reserve the marinade.
To make the sauce: Heat the reserved marinade in a small saucepan over medium heat until lightly bubbling, then stir in the cayenne pepper and whisk in the peanut butter and ½ cup water. Cook until hot and thick but still pourable, adding a little more water if needed. Remove from the heat and cover to keep warm.
Thread three slices of mushrooms lengthwise onto a skewer and repeat with the remaining mushrooms. Thread the gigante beans on separate skewers.
Heat a griddle, grill pan, or large cast-iron skillet over medium-high heat. Brush with a little oil. Cook as many mushroom skewers as will fit until softened and browned, about 4 minutes per side (pressing with a spatula if needed to help the mushrooms stay in contact with the pan). Repeat with the remaining mushroom skewers. Cook the bean skewers until heated through and lightly browned, about 2 minutes per side.
Transfer to a platter as you work and lightly tent the skewers with foil to keep them warm.
To serve: Pass the skewers and set out bowls of the sauce and coconut and let your guests dip and sprinkle as they see fit.