Whirl shallots, 2 slices ginger, and 1 garlic clove in a food processor until well chopped. Heat vegetable oil in a 5- to 6-qt. dutch oven over medium heat. Add shallot mixture and cook, stirring, until garlic begins to brown, about 2 minutes. Add rice and stir until well coated in oil.
Stir in broth and place chicken on top of rice. Cover and cook, without stirring, until chicken is cooked through and rice is tender, 15 to 20 minutes.
Meanwhile, purée green onions, remaining ginger and garlic, the cilantro, lime juice, green jalapeño, soy sauce, sesame oil, and 2 tbsp. water in a blender. Transfer to a bowl. Cut limes into wedges.
Lift chicken from pot and slice. Transfer rice to a platter and arrange chicken on top. Serve with limes, sliced red jalapeño and cucumber, and cilantro-ginger sauce.
Note: Nutritional analysis is per serving, with 2 tbsp. sauce