Preheat a gas or charcoal grill. (If you don’t have a grill you can make these on a skillet, griddle, or grill pan).
In a large bowl toss together carrots with ginger, olive oil, and a pinch of salt and pepper. Place on preheated grill in an even layer. In the now empty bowl whisk together coconut milk with zest and juice of 1/2 a lime. Set aside.
To a medium bowl add ground beef, Serrano pepper, lemongrass paste, 1/2 tsp salt, and a pinch of pepper. Gently mix together with your hands and form into 2 patties. Make a small dimple in the middle of each patty with your thumb to minimize shrinkage when cooking. Place on grill and flip carrots.
Spread mayonnaise on buns. After 3-5 minutes (depending on desired doneness) flip burgers and place buns on grill Mayo side down, preferably in a less hot section. Cook for another 3-5 minutes.
When burgers are done remove from grill along with the buns and transfer carrots to the bowl with coconut-lime dressing. Toss carrots with dressing and divide amongst plates. Assemble burgers with bun, red onion, burger patty, and cilantro leaves.