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 ¼ to ½ inch thick slices (~ 1 cm thick). Transfer the peaches to a medium or large bowl. Add the blueberries, brown 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. Toast the butter. Heat a 12-inch cast iron skillet 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.NOTE: If the butter doesn’t turn golden but still smells nutty in aroma, that’s fine. See Note 4 if you don't have a cast iron skillet. Make the cake batter. In a medium or large bowl, combine the flour, oats, cane sugar, salt, and baking powder. Whisk well to combine. Add in the oat milk and vanilla extract and almond extract (if using) and fold with a silicone spatula until combined.
Using a ladle or measuring cup, ladle the batter on top of the brown butter in the pan in different spots (ladling, instead of pouring all of the batter on top at once, helps the butter swirl and mix into batter). Arrange the peaches on top, then the blueberries, and spoon on the reserved juices.
Bake the cobbler for 45 to 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 to 20 minutes, then serve warm. If desired, scoop some vegan vanilla ice cream on each slice before serving.