Heat oil in a large nonstick skillet over medium-high heat. Add carrot and next 4 ingredients; saute until crisp-tender. Remove from heat. Add rice and next 3 ingredients; stir well.
Mound 1/4 cup rice mixture in center of each egg roll wrapper. Fold one corner of wrapper over filling, then fold left and right corners over filling. Push filling toward center of wrapper. Lightly brush exposed corner of wrapper with water. Tightly roll filled end of wrapper toward exposed corner; gently press corner to seal securely.
Place egg rolls on a large baking sheet coated with cooking spray. Coat tops of egg rolls with cooking spray. Bake at 350° for 20 minutes or until lightly browned. Serve warm. If desired, serve with low-sodium soy sauce or hot mustard.