Place garlic cloves on a piece of aluminum foil, and drizzle with 1 teaspoon oil. Wrap garlic in foil; roast until cloves are golden brown and very soft, about 30 minutes.
Meanwhile, heat 1 tablespoon oil in a large nonstick skillet over medium. Add onion, celery, lemongrass, and 1/8 teaspoon salt; cook, stirring occasionally, until onion is soft and translucent, about 4 minutes. Add ginger and chiles; cook, stirring occasionally, for 1 minute. Scrape onion mixture into a large bowl, and spread into an even layer. Let cool for 5 minutes. Wipe skillet clean, and set aside.
Add salmon, egg, panko, coriander, 1/4 cup mayonnaise, and 3/4 teaspoon salt to onion mixture in bowl; stir until well combined. Cover bowl using plastic wrap; refrigerate until cold, at least 1 hour or up to 12 hours.
While salmon mixture is chilling, squeeze roasted garlic cloves in a small bowl; discard peels. Mash garlic to equal about 3/4 tablespoon. Whisk in scallion, remaining 1/3 cup mayonnaise, and remaining 1/8 teaspoon salt. Cover aioli using plastic wrap, and refrigerate until ready to use.
Form salmon mixture into 8 patties (about 2 inches wide and 3/4 inch thick).
Heat remaining 2 tablespoons oil in reserved skillet over medium. Add salmon cakes to pan, and cook, flipping once, until browned on both sides and an instant-read thermometer registers 145°F, about 3 minutes per side.
Stir basil into aioli, and serve with salmon cakes.
