Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Step 2: In a medium bowl, combine the well-drained crushed pineapple, 1/4 cup granulated sugar, and cornstarch. Stir until well combined. Spread this mixture evenly into the prepared baking dish.
Step 3: In another bowl, beat the softened cream cheese and 1/2 cup granulated sugar with an electric mixer until smooth and creamy. Beat in the egg and vanilla extract until just combined. Gently spoon dollops of the cream cheese mixture over the pineapple layer.
Step 4: For the cobbler topping, in a large bowl, whisk together the flour, 1/2 cup granulated sugar, baking powder, and salt.
Step 5: Cut the cold butter into the dry ingredients using a pastry blender, two knives, or your fingertips, until the mixture resembles coarse crumbs.
Step 6: Pour in the milk and mix gently until just combined. The dough will be shaggy. Drop spoonfuls of this cobbler topping evenly over the cream cheese layer.
Step 7: Bake for 35-40 minutes, or until the topping is golden brown and the filling is bubbly. A wooden skewer inserted into the topping should come out clean. If the topping browns too quickly, you can loosely tent it with aluminum foil.
Step 8: Let the cobbler cool for at least 15-20 minutes before serving. This allows the filling to set slightly. Serve warm, optionally with a scoop of vanilla ice cream or a dollop of whipped cream.