In a small bowl , combine the turmeric, ginger, coriander, cayenne, and 1 teaspoon salt. Rub all over the chicken legs. Cover and refrigerate for at least 2 hours but preferably overnight.
Prepare a gas grill for direct grilling over medium heat. Grill the legs until they begin to brown on both sides, 3 to 5 minutes per side (watch carefully for flare-ups). Set aside.
Prepare the grill for indirect grilling. In an 8-quart heavy-duty pot, heat the oil over medium heat. Add the scallions and stir for 1 minute. Add the lemongrass and stir for 1 minute. Add the garlic and ginger and stir for 1 minute. Stir in the curry paste until the vegetables are evenly coated. Add the broth, fish sauce, and sugar and bring to a boil. Remove from the heat.
Nestle the chicken legs into the braising liquid. Set the pot on the grill over the cool zone. Cover the pot, close the grill lid, and cook until the legs are tender, about 30 minutes. Transfer the chicken to a tray.
Pour the braising liquid into a heatproof vessel, such as a Pyrex measuring cup , and let sit until the fat rises to the top. Skim off and discard the fat.
Prepare the grill for direct grilling over medium heat. Strain enough of the braising liquid to yield 1 cup and boil over medium heat in a small saucepan until reduced to 1/4 cup glaze, about 15 minutes. (Return the strained solids to the remaining cooking liquid).
Brush some of the glaze over one side of each chicken leg and grill glazed side down until the glaze begins to color, 2 to 3 minutes. Brush the other side of the legs with glaze and flip them over--the skin may stick to the grill a bit, so gently pry up any stuck areas before you flip. Grill until browned on the second side, 2 to 3 minutes.
To serve, reheat the remaining cooking liquid if necessary and season to taste with salt. Put some rice in each of 6 serving bowls, lean a leg against the rice, and ladle in some of the cooking liquid.
You can make this dish through the braising step the day before serving. Just cool and wrap the chicken and braising liquid separately. Refrigerate overnight and glaze the next day. To spread out the work even more, you can season the chicken the day before you braise.
To use a charcoal grill: To sear the chicken, build a medium-heat fire: Ignite about 5 quarts of charcoal (80 to 100 briquettes), using either a chimney starter or an electric starter. When the charcoal is burning well, spread it out over the surface of the charcoal grate and put the cooking grate in place. Let the charcoal burn down until it's coated with gray ash. To test the temperature, hold your hand about two inches above the cooking grate; when you can hold your hand there for 3 to 4 seconds (a medium-heat fire), you're ready to sear the chicken.
After searing, divide the coals evenly, banking them against two sides of the grill (use long-handled tongs to move the coals). Put the cooking grate in place; if your cooking grate has hinged sections, position them over the charcoal. Place an oven thermometer on the grate over the cooler area, close the lid, and let the coals burn until the thermometer reads about 350 degrees F. When ready, place the pot over the cool zone and braise as directed above. Check the thermometer every 20 minutes, replenishing the charcoal as necessary to keep the temperature between 325 degrees F and 375 degrees F. If the coals are still burning well, you can simply add a handful of unlit coals on top. Otherwise, you'll need to add lit coals.
To glaze the chicken, build a second medium-heat fire just as you did in the searing step. Continue with the glazing, as directed in the recipe.