Rinse rice in a strainer until water runs clear. Combine rice with 1 1/2 cups water in a saucepan, bring to boil, reduce heat to low, cover, and simmer until rice is tender, about 12-14 minutes. Spread cooked rice onto a baking sheet to cool, then season with 1 tablespoon sesame oil and salt.

Pour beaten eggs into a skillet over medium-high heat and cook into a flat layer without stirring. Remove from skillet and set aside to cool.

Separate nori sheets onto a flat surface, divide cooled rice between them, leaving 1/2-inch visible strip of seaweed at the top of each sheet. Arrange egg, cucumber, carrot, cheese, and ham strips over rice.