In a small saucepan, bring sake and mirin to a boil and boil 30 seconds. Lower heat to medium and add miso, whisking until dissolved. Add sugar and ginger and whisk until sugar dissolves. Pour half of miso mixture into a large bowl and the other half into a medium baking dish and let cool, about 10 minutes.
Slice eggplant on the diagonal into 1/4-in.-thick ovals and add to miso mixture in bowl, turning to coat. Nestle salmon fillets, skin side up, into miso mixture in baking dish to coat all the flesh, but not the skin. Cover both bowl and dish with plastic wrap and refrigerate at least 8 hours and up to overnight.
Prepare a gas or charcoal grill for indirect heat. If using a gas grill, turn all burners to high and close lid. When the temperature inside the grill reaches 400°, lift lid and turn off one of the burners, creating the indirect-heat area. If using a charcoal grill, light 50 to 60 briquets and let burn until just covered with ash, 20 to 30 minutes. Mound them to one side, leaving a cleared area for indirect cooking.
Shake marinade off eggplant. Oil an unperforated sheet of heavy-duty foil with canola oil and arrange eggplant on it in 1 layer. Drizzle with 1/3 cup oil and grill over indirect heat area, 15 to 20 minutes, or until soft. Set aside.
Fold two 12- by 18-in. pieces of heavy-duty aluminum foil in half widthwise to form rectangles. Using the tip of a small knife, make holes in rectangles about 2 in. apart and widen each hole to the size of a dime. Grease foil with oil; set over direct heat for 2 minutes.
With paper towels, wipe marinade off salmon. Brush salmon skins with oil and set 2 fillets, skin side down, on each foil rectangle. Cover grill (if using charcoal, open vents on lid) and cook fish until skin is light brown and really sizzling, 5 to 6 minutes. Using tongs, slide foil to indirect heat, cover, and cook until all but top 1/4 in. is cooked, 3 to 8 minutes. Slide fish back over direct heat, cover, and cook until fish is cooked through (cut to test) and skin is browned and crisp, about 3 minutes. While salmon is finishing on direct heat, reheat eggplant over indirect heat.
Transfer foil with salmon to a rimless baking sheet and, sliding an offset cake spatula or other thin spatula between salmon skin and foil, very gently free fish from foil. Season each fillet with a few drops of lemon juice. Serve salmon and eggplant sprinkled with chives.
Note: Nutritional analysis is per serving.
* Percent Daily Values are based on a 2,000 calorie diet. Your daily value may be higher or lower depending on your calorie needs.