Bring water to a boil in a heavy, large saucepan then stir in the oats. Reduce heat to a simmer then cook oats until they begin to thicken, about 10 minutes; mix in the apple, carrot, sultanas, cinnamon, nutmeg, ginger and salt. Let the oats simmer until tender, about 20 more minutes.
While the oats are simmering, melt butter in a frying pan over medium-low heat then 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 yoghurt.