Preheat your oven to 350°F and grease a 9x13 inch baking dish (or similar 2-quart dish) with vegan butter or oil. This ensures the cobbler won't stick and gives you time to prep while the oven heats.
Slice the peaches into 1/2-inch thick wedges and place them in the prepared baking dish. In a small bowl, whisk together the 1/4 cup sugar, cornstarch, and lemon juice, then pour this mixture over the peaches and gently toss to coat evenly. The cornstarch will thicken the peach juices as they cook, creating a lovely filling. I like to let the peaches sit for just a minute so they start releasing their juices.
In a medium bowl, whisk together the flour, baking powder, salt, and cinnamon. In a separate bowl, combine the softened vegan butter, 3/4 cup sugar, vanilla, and soy milk, mixing until smooth and well combined. I find that slightly softened butter (not melted) creates a better texture—it gives you a topping that's tender but holds together nicely.
Add the wet ingredient mixture from Step 3 to the dry ingredient mixture from Step 3, stirring just until combined—don't overmix, as this keeps the topping tender. Drop spoonfuls of the batter evenly over the peaches, leaving some gaps so steam can escape and the peaches show through slightly. The topping will spread as it bakes.
Bake for 30-35 minutes, until the topping is golden brown and a toothpick inserted into it comes out clean. The filling should be bubbling gently around the edges. Remove from the oven and let cool for 5-10 minutes before serving—this allows the filling to set slightly so it won't run all over the plate.