Place large skillet on stovetop over medium-high heat. Add 1 ½ teaspoons sesame oil and 3 teaspoons avocado oil to skillet and continue heating pan until oil is hot and shimmery, swirling and tilting pan as needed to distribute oil across entire surface of pan.
When oil is hot, place 12 ounces medium-large shrimp in skillet. Let shrimp cook undisturbed 1 minute.
After 1 minute, flip shrimp over. Quickly add 2 tablespoons soy sauce, 1 tablespoon unsalted butter, 2 teaspoons fresh lemon juice, salt, and freshly cracked white pepper to skillet.
Cook shrimp 1 to 2 minutes, stirring ingredients only once or twice, until shrimp is opaque, pink color and curled into loose "C" shape.
When shrimp are cooked-through, transfer shrimp to medium bowl and set aside. Wipe skillet out with paper towels, then return skillet to heat.
Add 1 teaspoon avocado oil to skillet. Heat skillet over medium-high heat, swirling and tilting pan occasionally, until oil is hot and shimmery.
When oil is hot, add ½ cup diced white onion and 1 cup frozen mixed vegetables to skillet. Sauté ingredients, stirring often, approximately 3 minutes or until onions are almost completely translucent.
Once onions are almost translucent, push vegetables to one side of skillet. Crack 2 large eggs directly into empty side of skillet. Scramble eggs with spatula until eggs are cooked-through.
Once eggs are cooked-through, add 4 cups cooked, cooled rice and 3 tablespoons unsalted butter to skillet.
Stir to incorporate all ingredients, then cook mixture 5 minutes, stirring frequently.
After 5 minutes, add 3 tablespoons soy sauce to skillet and stir to incorporate. Continue cooking mixture 1 minute more.
Taste rice and season with salt and freshly cracked white pepper as desired.
Return shrimp to skillet and stir to incorporate. Cook mixture just until all ingredients are warm. Be careful not to overcook shrimp.
Portion shrimp fried rice into serving bowls. If desired, garnish with white sesame seeds and thinly sliced green onions, then serve fried rice warm and enjoy.