Grease a 9-inch baking dish with cooking spray and set aside. In the bowl of a food processor pulse dates and sugar until mixture is rough textured and grainy.
Add eggs and Chobani and pulse to combine. Add water and vanilla and pulse to combine then with processor running add 8 T melted butter.
In a medium bowl mix flour, baking powder, baking soda and salt. Add to Chobani mixture and pulse to combine.
Pour into the prepared baking dish and bake at 350°F until set and no longer soft in the center, about 30 minutes.
Stir brown sugar into remaining 4 T melted butter in a saucepan. Once sugar is melted add rum and pour over still-warm cake, spreading it evenly so it melts across the surface. Cool, cut into 9 squares and serve.