Add one tablespoon of butter to a small skillet over medium heat. Pour the beaten eggs and gently scramble until set, about 3 minutes. Set aside.
In a large skillet, heat sesame oil over medium heat. Add the garlic and cook for 30 seconds, until fragrant. Add the shrimp and cook until pink and opaque, about 4 minutes. Remove the shrimp from the skillet and set aside.
Melt the remaining tablespoon of butter in the same skillet over medium heat. Add the frozen peas and carrots. Cook, stirring often for 6 to 8 minutes, until heated through and tender.
Add the rice, soy sauce, rice vinegar, and white pepper to the skillet. Use a spatula to break up any clumps of rice. Increase the heat to medium-high and stir-fry for 4 minutes.
Add the cooked shrimp and scrambled eggs to the skillet. Toss everything together and cook until heated, about 2 to 3 minutes.
Remove from the heat and stir in the green onions—season with salt and pepper to taste. Sprinkle with sesame seeds, if using. Serve hot.