In a small bowl (preferably with a spout), beat together the eggs and salt until smooth. To make the color and texture more uniform, pour the eggs through a fine mesh strainer if desired. Mix the carrots and scallions into the eggs.
Place a Tamagoyaki omelette pan or medium (8 inch) nonstick skillet over medium low heat. Grease with a layer of oil.
Once thoroughly heated, pour 1/3rd of the egg mixture into the pan in a thin, even layer. Gently cook until the bottom is set, but the top is still slightly wet.
Use a spatula or chopsticks to gently lift one side (short side if using a rectangular pan) and fold it over about an inch. Continue to lift and fold/roll until the egg reaches the other side.
Grease the exposed part of the pan with oil and pour in another 1/3rd of the egg mixture in a thin, even layer. Once the bottom is set and the top is still a little wet, tightly roll the egg back to the opposite side.
Repeat the process one more time with the remaining 1/3rd egg, rolling again to the opposite side to form one log with multiple layers of egg.
Remove the roll to a plate and allow to rest for 5 minutes. Slice the roll into even pieces about 3/4-1 inch thick.