Have ready 1/2 cup water. If using Chinese brown sugar, put it and 2 Tbs. water in a 10-inch straight-sided skillet over medium heat. Stir frequently with a wooden spoon, chopping up the sugar with the spoon until it dissolves completely, 4 to 5 minutes. (If using white sugar, put it—without the water—in a 10-inch straight-sided skillet over medium. Cook until it starts to melt at the edges and turn golden brown, about 5 minutes.)
Reduce the heat to medium low and continue cooking, stirring constantly if using brown sugar or gently swirling the pan if using white sugar, until the bubbling caramel darkens to a reddish brown, 3 to 4 minutes. Remove the pan from the heat and, with your face averted to avoid steam and spatters, carefully pour the 1/2 cup water into the pan. The caramel may harden; if it does, set the pan over medium-high heat and stir until it dissolves. Stir the liquid to blend in the caramel and pour it into a heatproof measuring cup or bowl.
Wipe out the pan and heat the olive oil over medium high. Add the shallot and ginger and cook, stirring frequently, until they’re softened and starting to brown, 2 to 3 minutes. Transfer to a bowl.
Set the skillet back over high heat. Add the seasoned chicken pieces and cook until they lose their raw color on the outside, about 2 minutes per side; the pan will be crowded and the chicken needn’t brown. Stir in the fish sauce, salt, pepper, chile flakes, and reserved caramel. Reduce the heat to medium and cook at a vigorous simmer, turning the chicken every few minutes, until the chicken is cooked through (cut into a piece to check), about 20 minutes. Stir in the reserved ginger and shallot and cook for 3 to 4 minutes to blend the flavors. Transfer to a serving plate and garnish with the scallion rings.
Serve with jasmine rice and sautéed broccoli.