Rinse turkey thighs and pat dry.
In a bowl, mix basil, cilantro, mint, garlic, brown sugar, soy sauce, ginger, fish sauce, oil, and chili paste. Spread about 2 tablespoons of the mixture evenly over boned side of 1 turkey thigh. Set remaining thigh, boned side down, over herb mixture, aligning with first thigh. Tie thighs together at 1-inch intervals with cotton string to create a cylinder about 3 inches wide and 7 inches long. With your fingers, pat remaining herb mixture evenly all over roast.
Prepare barbecue for indirect heat.
If using charcoal briquets, mound and ignite 60 briquets on the firegrate of a barbecue with a lid. When briquets are dotted with gray ash, in 15 to 20 minutes, push equal amounts to opposite sides of firegrate. Add 5 more briquets to each mound of coals now and after 30 minutes, if grilling takes longer than that.
If using a gas barbecue, turn all the burners to high, close lid, and heat for 10 minutes. Then adjust burners for indirect cooking (heat on sides of grill not down center under food) and keep on high.
Set a drip pan on firegrate between coals or burners. Set barbecue grill in place. Set roast in center of grill, not over heat. Cover barbecue; open vents for charcoal.
Cook roast until a thermometer inserted in center of meat reaches 170°, 40 to 45 minutes.
Transfer roast to a platter and, keeping it warm, let rest 5 to 10 minutes. Garnish platter with lime wedges. Cut roast into 1/2-inch-thick slices to serve. Squeeze juice from lime wedges over portions and add salt to taste.