Preheat oven to 350 degrees F.
Put a large pan on high heat. Add a good layer of oil (a little more than just coating the pan). When the oil is hot enough, add the chicken. You should hear a sizzle as the meat hits the pan. Let the chicken pieces brown for about 5 to 6 minutes on one side. Then gently flip them over, and place the pan in the oven for about 10 more minutes of roasting, to make sure the chicken pieces are cooked through.
When the chicken comes out of oven, remove the chicken from pan, and set aside.
Put the pan over medium heat, and pour the ginger beer into the pan to deglaze it, stirring up all the brown bits. Add the cinnamon and Kaffir lime leaves, and let simmer for a couple of minutes, and then add the butter, and stir in.
Toss the chicken pieces with the vegetables, and the juice of a lime. Plate up in a mound, and drizzle the sauce over top, and place 2 or 3 fried dumplings on the side. Add a couple of lime wedges, for garnish.
In a large bowl, mix together the flour, cornmeal, baking powder, baking soda, salt, and sugar. Work in the cold butter with your hands, rubbing the butter bits into the flour, breaking them up and aiming for small pea-size pieces.
Add the milk to make a loose dough that will barely hold its shape without running off a spoon. Drop by large spoonful into the hot oil, and fry about 3 minutes per side, or until nicely browned but not too dark. When you bite into it the crust should be crunchy but the inside soft and tender.
Serve the warm dumplings beside the Jerk Chicken.