Pat shrimp dry with paper towel. Chop shrimp into small pieces. Use a cleaver or a heavy knife to chop and smash the shrimp into a paste. Alternatively, you could make the shrimp paste with a food processor. Transfer to a big bowl.
Add egg white, coconut oil, salt, potato starch, and white pepper powder to the shrimp paste. Chop 2 slices fresh ginger. Use a garlic press to squeeze about 1/4 teaspoon ginger juice onto the shrimp paste (or add ginger powder).
Mix the shrimp and all the ingredients until it forms a consistent and sticky paste that can just hold its shape.
Mix the all the ingredients for the sauce with 1/4 cup water in a small bowl.
Add oil to a wok until it’s 1 inch (2.5 cm) deep (*see footnote) and heat over medium high heat.
When heating the oil, prep the vegetables.
When the oil is getting hot (registering 350 degrees F / 150 C), use a spoon to carefully shape the shrimp paste into a ball. Carefully place it into the oil. Continue to form and place them into the oil one by one. Cook a few shrimp balls at a time, so the wok won’t become too crowded, and the oil temperature doesn’t drop too fast.
Fry until the shrimp balls turn pale golden brown, 3 to 4 minutes. Use a slotted spoon to drain them gently, then transfer them to a plate lined with paper towels.
Heat 1/2 tablespoon oil in a nonstick skillet over medium high heat until warm. Add carrot and broccolini. Stir and cook for 1 minute. Swirl in 2 tablespoons water and cover immediately. Steam until the veggies are cooked through, 2 minutes. Transfer to a plate.
Heat a skillet over medium low heat until warm. Add lemon sauce. Cook and stir until sauce thickens. Turn off heat.
(Option 1) Add shrimp balls and veggies back into the skillet. Stir until the ingredients are evenly covered with sauce. Transfer everything to a plate. Grate black pepper over the dish to garnish. Serve by itself or over steamed rice as a main.
(Option 2) Transfer the sauce to a bowl to cool. Add shrimp balls and veggies to a big bowl, drizzle with the lemon sauce to serve as a cold salad.