Bring water to a boil in a heavy, large saucepan, and stir in the oats. Reduce heat to a simmer, and cook oats until they begin to thicken, about 10 minutes; mix in the apple, carrot, raisins, cinnamon, nutmeg, ginger, and salt. Let the oats simmer until tender, about 20 more minutes.
While the oats are simmering, melt butter in a skillet over medium-low heat, and stir in the pecans. Toast the nuts until fragrant and lightly browned, 2 to 5 minutes, then sprinkle with brown sugar and stir until sugar has melted and coated the pecans.
Serve in bowls, each topped with about 2 tablespoons of the pecan mixture and a dollop of yogurt.
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