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Finely chop the scallion and carrot.
Crack and beat the eggs, in a bowl with a spoon, a fork, or a pair of chopsticks until the yolks and whites are blended well with no visible strings of whites. Stir in salt and pepper.
Stir in the chopped vegetables.
Add a teaspoon of vegetable oil to a medium size non-stick pan and distribute the oil evenly using a napkin or paper towel. Preheat over medium low heat, until it feels hot when you place your hand 2 to 3 inches above the bottom of the pan.
Add 1/2 of the egg mixture to the pan. Swirl it around to cover the pan. Let it cook until the top begins to set but is still wet.
Using a spatula, lift one end of the egg (about 1-1/2 inches) and fold it over to the other side. Reduce the heat or raise the pan away from the heat for a little while if the bottom is browning.
Lift the folded part and fold it again. Pull the egg to the rolled side, making room on the round end for more egg mixture. Add 1/2 of the remaining egg and spread to cover the open space.
When the egg addition is set, lift the folded part and fold it again. Pull the egg to the rolled side and repeat the process with the remaining egg to make an about 2-inch egg log.
Remove from the pan and let cool for about 5 minutes. Slice the egg into thick, even pieces (3/4 to 1-inch thick).