Bring water to a boil in a large pot. Add chicken, 1 cup ginger, 1 head garlic, sugar, and salt to the pot. Return to a boil; simmer for 35 minutes. Transfer chicken to a plate, cover to keep warm, and set aside.

Heat coconut oil in a rice cooker or heavy-bottomed pot. Add 4 pieces chopped ginger, pandan leaves, 1 tablespoon garlic, and shallots; cook and stir until golden and aromatic.

Add rice to rice cooker or pot; stir to coat with oil. Stir in chicken broth; set rice cooker cycle according to manufacturer's instructions. If using a pot, bring chicken broth to a boil, cover, reduce heat to low, and simmer 15 minutes.

Meanwhile, combine Thai chiles, pickled garlic, fermented soybeans, 1/3 cup ginger, vinegar, and soy sauce in the bowl of a food processor or blender; pulse until liquefied but not smooth.

Debone chicken; discard bones. Cut meat into 1-inch pieces.

Place chicken pieces over cooked rice; top with Thai chile sauce or serve sauce on the side. Garnish with cilantro and cucumbers.