Step 1: Preheat oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish.
Step 2: In a large bowl, combine the thoroughly drained crushed pineapple, 1 cup of granulated sugar, and vanilla extract. Stir well and set aside.
Step 3: In a separate medium bowl, beat the softened cream cheese with the remaining 1/2 cup of granulated sugar until smooth and creamy. Ensure there are no lumps.
Step 4: Pour the pineapple mixture into the prepared baking dish, spreading it evenly.
Step 5: Spoon small dollops of the cream cheese mixture over the pineapple layer. Do not spread; the dollops will melt and create creamy pockets.
Step 6: In another medium bowl, whisk together the self-rising flour and milk until just combined. Do not overmix; a few lumps are acceptable.
Step 7: Pour the melted butter into the flour and milk mixture and stir gently until just incorporated.
Step 8: Carefully spoon the cobbler topping over the pineapple and cream cheese mixture in the baking dish.
Step 9: In a small bowl, mix together the 1 tablespoon of granulated sugar and 1/2 teaspoon of ground cinnamon (if using). Sprinkle this mixture evenly over the top of the cobbler topping.
Step 10: Bake for 40-50 minutes, or until the topping is golden brown and the filling is bubbly. If the top browns too quickly, you can loosely tent it with aluminum foil.
Step 11: Let the cobbler cool for at least 15-20 minutes before serving. This allows the filling to set. Serve warm, ideally with vanilla ice cream or whipped cream.