Lightly grease an 8x8" baking dish or similar-sized casserole dish with oil or butter. Move an oven rack to the middle position and heat the oven to 325°F.
Whisk together ½ cup plain or vanilla Greek yogurt, ½ cup granulated sugar, 1 large egg, 1 Tablespoon cornstarch, and 1 teaspoon vanilla extract in a large bowl. Stir in 4 cups chopped peeled peaches. Pour this into the prepared baking dish and set aside.
In a separate medium or large bowl, whisk together 1 cup all-purpose flour, ⅓ cup granulated sugar, 1 teaspoon baking powder, ¼ teaspoon salt, and ¼ teaspoon baking soda.
In a small bowl or large measuring cup, stir together ½ cup plain or vanilla Greek yogurt, 1 large egg, 2 Tablespoons unsalted butter, and ½ teaspoon vanilla extract. Pour the yogurt mixture into the flour mixture and fold it together with a spatula until it forms a stiff, dough-like batter.
Drop the batter in small portions evenly all over the peach filling mixture in the baking dish. The easiest tool for this is a portion scoop (I used a #40 size), but you can also use two spoons (one to scoop batter, and the other to scrape the batter from the first spoon). There is no need to spread the batter, it will spread during baking, forming that cobblestone-like appearance that cobblers are known for.
Transfer the baking pan to the oven and bake for 40-50 minutes, until the topping is golden brown. Note that if your peaches are cold/frozen, and/or you are using a glass baking dish vs a metal one, you will likely need to aim for the higher end of the time (I baked my fresh peach filling in a glass pan for 50 minutes). Allow the cobbler to cool for at least 20 minutes before serving.
To serve, scoop portions from the baking pan (make sure each portion includes both filling and topping) and serve with ice cream or whipped cream if desired.
Cover leftovers with plastic wrap or a lid and refrigerate for up to 2 days. Reheat cold cobbler in the microwave if desired.