Heat a grill or grill pan to medium-high and brush grate with oil. In a small bowl, combine 1 tbsp of the brown sugar, 1/4 tsp of the salt and 1/8 tsp of the cayenne. Rub onto chicken breasts. Grill chicken over medium-high heat 5 to 6 minutes per side, until cooked through.
Toss peppers and scallions in 1 tsp of the vegetable oil. Skewer peppers and grill 4 minutes; flip and grill another 4 minutes. Grill scallions 2 minutes per side. Thinly slice chicken on the bias. Cut peppers into rings, discarding stems, and chop scallions.
Bring a pot of lightly salted water to a boil. Add soba and cook 4 to 5 minutes, until tender. Drain and rinse immediately under cold water.
Heat a skillet over medium heat. Toast coconut 2 to 3 minutes, stirring frequently. Remove to a plate. Add remaining 1 tsp oil to skillet. Add garlic and ginger to skillet; sauté 1 minute. Pour in coconut milk, lime juice, fish sauce, remaining 1 tbsp brown sugar, remaining 1/4 tsp salt and remaining 1/8 tsp cayenne. Bring to a simmer. Mix cornstarch with 1 tsp cold water. Add to mixture and simmer 2 minutes, until thickened.
In a large bowl, toss sauce with chicken, peppers, scallions and soba. Divide among 6 bowls and garnish with coconut and cilantro.