Place the rice in a fine-mesh sieve and rinse under cold water until the water runs clear. Transfer to a medium saucepan, adding 1 teaspoon of salt and 3 cups of water.
Bring to a boil, then cover and reduce the heat to the lowest setting. Simmer, undisturbed, for 15 minutes. Then, remove from the heat and let it stand for at least 15 minutes. Uncover and fluff the rice to break down the clumps. Let it cool completely, then transfer to a baking sheet, cover with plastic and refrigerate overnight.
The next day, mix the softened butter, lemon juice, 1 teaspoon soy sauce and garlic cloves in a bowl. Stir vigorously until smooth. Reserve.
Season the chicken breasts with salt. Heat 1 tablespoon of vegetable oil in a large non-stick skillet, over medium-high heat, and brown the chicken breasts on all sides, about 2 minutes per side.
Transfer the chicken breast to a cutting board and cut into cubes. Return the cubes to the skillet and continue cooking until golden brown, about 3-5 more minutes. Add 1 tablespoon of the garlic butter and mix to combine. Remove the chicken cubes to a plate and reserve.
Return the skillet to the heat, adding another tablespoon of oil. Add the chopped onion and carrots, sautéing until softened, about 5-8 minutes. Then, add the green onions and continue cooking for another couple minutes. Remove and reserve.
Heat more oil in the skillet and add the eggs, scrambling them, about 1-2 minutes. Remove and reserve.
Place a large wok pan on the stove and heat on medium-high. Add enough oil to coat the whole pan. Add the cooked rice and let it cook for about 5 minutes, or until warmed through. Then, add the remaining garlic butter and toss to combine, making sure all the rice gets coated with the butter.
Add the reserved veggies and mix until combined. Then, carefully stir in the reserved chicken and eggs. Stir-fry for another 2-3 minutes or until the rice begins to brown slightly.
Season with salt and pepper, and drizzle soy sauce to taste.
Serve, garnished with toasted sesame seeds.