In a medium bowl, combine the minced lemongrass, fish sauce, soy sauce, brown sugar, lime juice, chili paste, garlic, and vegetable oil. Whisk until well combined to create the marinade.
Add the shrimp to the marinade, ensuring they are well coated. Cover the bowl with plastic wrap and refrigerate for at least 30 minutes, or up to 2 hours for more flavor.
Preheat your grill to medium-high heat. If using wooden skewers, soak them in water for 30 minutes to prevent burning.
Thread the marinated shrimp onto skewers, leaving a little space between each shrimp for even cooking.
Place the skewers on the grill and cook for 2-3 minutes on each side, or until the shrimp are opaque and cooked through.
Remove the shrimp from the grill and let them rest for a couple of minutes.
Serve the grilled shrimp garnished with fresh cilantro and lime wedges on the side.