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 (see Test Kitchen Note). 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 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. Tip: Spread pecans on a baking sheet and bake at 350°F, stirring once, until fragrant, 7 to 9 minutes. Test Kitchen Note: Perfectly cooked rice is not simple. In fact, it's something that we struggle with occasionally in the Test Kitchen. To have the most success cooking whole-grain rice, we recommend using a pan with a tight-fitting lid, cooking on your coolest (or simmer) burner and making sure the rice is simmering at the "lowest bubble." While testing the recipes that use less than 1 cup of dry rice, we found that the cooking time varied greatly depending on what stove we used. Although whole-grain rice usually requires 50 minutes of cooking, we found smaller volumes of rice were sometimes done in as little as 30 minutes (and burned at 50 minutes). So, when cooking a small batch of rice, start checking it after 30 minutes to make sure it doesn't burn.