Heat vegetable oil in a large skillet over medium heat.
Add minced garlic and thinly sliced shallots; sauté until fragrant and translucent.
Stir in red Thai curry paste and cook for 1-2 minutes to release aroma.
Pour in coconut milk, then add fish sauce, brown sugar, fresh lime juice, and torn kaffir lime leaves. Stir well and bring to a gentle simmer.
Add sliced red bell pepper and halved baby eggplants to the sauce; simmer for about 5 minutes until vegetables are crisp-tender.
Gently place salmon fillet portions into the curry sauce, partially submerging them.
Cover and simmer for 8-10 minutes or until salmon is tender and flakes easily.
Remove from heat and stir in fresh Thai basil leaves.
Serve each salmon portion partially submerged in curry sauce in a shallow white bowl with vegetables arranged evenly around.
Garnish with fresh cilantro leaves on top before serving.