- Bring rice and water to a boil in a medium saucepan.
- Reduce heat to low, cover, and simmer at the lowest bubble until the water is absorbed and the rice is tender, 30 to 50 minutes.
- Remove from the heat and let stand, covered, for 10 minutes.
- Meanwhile, heat 2 teaspoons oil in a large skillet over medium heat.
- Add shallots; cook, stirring often, until soft, 2 to 3 minutes.
- Add shredded carrots, reduce the heat to low and cook, stirring often, until softened and the shallots are lightly browned, about 4 minutes. Remove from the heat.
- Preheat oven to 400°F.
- Transfer the cooked vegetables and rice to a large food processor. Add pecans, goat cheese, egg white, thyme, salt and pepper. Pulse until well blended but still a little coarse.
- Scrape into a large bowl. With wet hands, form the mixture into six 3-inch patties (about 1/2 cup each).
- Heat the remaining 2 teaspoons oil in a large nonstick skillet over medium heat. Add the patties and cook until well browned, 3 to 4 minutes per side.
- Transfer to a baking sheet and bake until an instant-read thermometer inserted into the middle registers at least 160°F, 10 to 15 minutes.
- Serve hot.