Bring water to a boil in a large, heavy saucepan; stir in oats. Reduce heat to a simmer; cook oats until they begin to thicken, about 10 minutes. Stir in apple, carrot, raisins, cinnamon, nutmeg, ginger, and salt; simmer until tender, about 20 minutes more.

Meanwhile, melt butter in a skillet over medium-low heat. Stir in pecans; cook until fragrant and lightly browned, 2 to 5 minutes. Sprinkle with brown sugar; stir until sugar has melted and coats pecans.

Top oatmeal servings with about 2 tablespoons pecan mixture and a dollop of yogurt.