Preheat your oven to 350°F. Place the 1 stick of unsalted butter directly into a 9×13-inch metal or glass baking dish, and slide the pan into the oven for 4 to 5 minutes just until the butter completely liquefies. Remove the dish and set it flat on the counter; do not drain the melted fat.

In a medium mixing bowl, whisk together the all-purpose flour, baking powder, salt, and 1 cup of granulated white sugar until evenly blended.

Pour the whole milk and vanilla extract into the dry flour mixture, switching to a silicone spatula to fold the ingredients together just until the flour lines disappear and a thick batter forms. Over-mixing at this stage develops tough gluten networks that can ruin the cobbler's tender texture.

Pour the cake batter gently and uniformly directly over the melted butter in the baking dish. Do not stir. Leaving the batter floating on top of the liquid fat allows the thermal currents in the oven to lift the crust through the topping cleanly.

In a medium saucepan set over medium-high heat, combine the packed brown sugar, ½ cup of granulated white sugar, corn syrup (or alternative syrup), water, pecan halves, and ground cinnamon. Stir continuously for 3 to 4 minutes until the sugar crystals fully dissolve and the mixture reaches a hard, rolling boil.

Remove the boiling pecan mixture from the burner and carefully ladle it in steady streams evenly across the top of the raw cake batter. Do not stir. Gravity will naturally pull the dense caramel down while forcing the cake batter to push upward around the nuts during the bake.

Slide the dish carefully onto the middle oven rack and bake uncovered at 350°F for 40 to 45 minutes. Watch for the cake edges to turn a deep golden brown and the center to rise completely.

Remove the cobbler from the oven and place it on a wire cooling rack to rest for 15 minutes before serving. This cooling window allows the internal molten sugars to stabilize into a thick, fudgy, and spoonable custard. Serve warm with a scoop of vanilla bean ice cream.