In a medium saucepan, heat olive oil over medium-low heat. Add onion and carrot; cook, stirring frequently, until the onion is very tender, 15 to 20 minutes. Add garlic and ginger; cook 2 minutes.
Add rice, stirring to coat. Add water and salt; bring to a boil. Reduce to a simmer; cover, and cook until the rice is tender, about 17 to 20 minutes.