For the lime aioli: Place the vegetable oil, mustard, limes leaves, egg yolk and lime juice in a tall container. Blend the ingredients using a hand blender until the lime aioli becomes thick. Place the mixture in a bowl, season with salt, cover and place in fridge.
For the chicken salad: Place the chicken, limes leaves, celery, onions, bay leaves, peppercorns, some salt and 10 cups cold water in a large pot and bring to a boil. Once the water comes to a boil, lower the heat to a simmer and skim the impurities. After about 1 hour, turn the heat off and allow the chicken to rest for 30 minutes.
Remove the chicken from the broth and remove the skin. Strain and reserve 3 3/4 cups chicken stock for the brown rice pilaf. Refrigerate or freeze the remaining stock. Once the chicken has cooled, shred the chicken meat by hand and place in a bowl. Mix the lime aioli with the shredded chicken meat, season with salt and place in the fridge until assembly.
For the crunchy lime topping: Place the celery, olive oil, green onions, lime zest and juice in a small bowl. Season with salt, mix and reserve for assembly.
For the charred tomato: In a cast-iron pan, heat a couple tablespoons olive oil over medium to high heat. Season the tomatoes with sugar, salt and pepper. Sear the tomatoes on both sides until charred, remove from the heat and reserve until assembly.
For the brown rice pilaf: Heat the vegetable oil in a wide, tall-rimmed pot over medium heat. Add the shallots and cook until tender and translucent. Add the bay leaf and rice and stir to coat the rice in oil. Add the reserved 3 3/4 cups chicken stock and the soy sauce and stir. Bring to a boil, then reduce the heat to a low simmer and place the lid ajar over the pot. Cook until rice is tender, 30 to 40 minutes, then fluff with a fork and set aside.
To assemble the lime leaf chicken salad: Place some brown rice pilaf on the bottom of each serving plate. Place 2 pieces of charred tomato on each plate over the pilaf. Place equal amounts of lime leaf chicken salad on top of the charred tomatoes. Spoon the crunchy lime topping over. Garnish each plate with celery leaves and drizzle of olive oil.
*Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell. For recipes that call for eggs that are raw or undercooked when the dish is served use shell eggs that have been treated to destroy Salmonella, by pasteurization or another approved method.