Arrange a rack in the middle of your oven and preheat it to 375°F/190°C.
Cut the peaches in half and remove the pits, then cut the peaches into 1/4-1/2 inch thick slices (3/4 cm - 1 1/4 cm). Transfer the peaches to a medium or large bowl. Add the blueberries, coconut sugar, cinnamon, nutmeg, ginger, and cardamom (if using), and toss gently to combine. Set aside for 30 minutes to allow the fruit to absorb the flavors.
Brown the butter. Heat a large cast iron skillet (11- or 12-inch) over medium heat. Add the vegan butter to the pan, and use a spatula to spread it across the sides of the pan. Once it’s melted, foamy, and at a bubble (it should take 2 to 3 minutes), heat for another 2 minutes, stirring frequently to prevent burning, then take the pan off the heat.See note below on alternative pan sizes.** Make the cake. In a medium or large bowl, combine the flour, oats, cane sugar, salt, and baking powder. Whisk in the oat milk and vanilla extract and almond extract (if using) until well combined. Using a ladle or measuring cup, ladle the batter on top of the brown butter in the pan. Ladling, instead of pouring all of the batter on top at once, helps the butter swirl and mix into batter. Top the cobbler with the peach-blueberry mixture.
Bake the cobbler for 45-50 minutes, rotating the pan 180° halfway through to ensure even browning, until the top is deeply golden brown and bubbling.
Transfer to a wire rack to cool for 10 minutes, then serve warm. If desired, scoop some vegan vanilla ice cream on each slice before serving.