Place rice and water in instant pot. Cook on RICE/RISOTTO setting for 12 minutes, then release pressure naturally 5 minutes. Fluff rice with a fork and transfer to a large bowl to cool
Heat 1 tablespoon oil in instant pot using SAUTE mode. Once heated, pour beaten eggs in bottom of instant pot. Use a spatula to gently lift up edges of eggs, gently folding until eggs are set. Fold eggs into quarters and carefully remove from instant pot to a cutting board. Finely chop eggs and set aside.
With instant pot still on SAUTE mode, saute chicken, onion, carrot, and garlic in remaining 1 tablespoon oil until chicken is no longer pink and veggies are just slightly tender. Add rice, soy sauce, vinegar, and sugar to instant pot and saute several minutes until rice is fragrant.
Add eggs back to fried rice and season with salt and pepper to taste. Enjoy!