Preheat the oven to 350°F (175°C). In a small bowl or measuring cup, whisk together the water and ground flaxseed and set aside to thicken for a few minutes.
In a large bowl, stir together the oats, baking powder, cinnamon, and salt. In small bowl, whisk together the nondairy milk, maple syrup, and flaxseed mixture. Add to the oat mixture and stir until well combined.
Heat the oil in a 10-or 12-inch (25- or 30-cm) cast-iron skillet over medium heat. Add the peaches and let them sizzle and cook for 1 minute. Sprinkle with 2 tablespoons of the sugar and the vanilla. Cook, stirring occasionally, for about 4 minutes, until the fruit is soft and just beginning to brown. Pour the oat mixture over the fruit, then stir gently until the fruit is evenly distributed.
Bake for 20 minutes, then sprinkle the remaining 1 tablespoon sugar and the nuts over the top. Bake for about 10 minutes longer, until the fruit is bubbling in places and the surface is firm to the touch. Let cool for 5 to 10 minutes before serving.