Preheat the oven to 350°F (175°C). Line an 8-inch by 8-inch (20cm by 20cm) square pan with parchment paper, leaving enough overhang to lift the cake out after baking. Alternatively, you can use a 9-inch (23cm) round pan.
In a saucepan, combine the dates and water. Bring to a boil, then remove from heat. Let the dates soak in the hot water for 10 minutes. After soaking, transfer the mixture to a blender, food processor, or use an immersion blender to blend until it resembles a thick paste. Set aside.
In a large bowl, combine the butter, white sugar, brown sugar, and vanilla extract. Beat with a hand mixer until fully combined and fluffy. Add the oil and beat until incorporated.
Add the eggs, one at a time, beating well after each addition.
Add the date paste to the mixture and beat until well combined.
Sift the flour, baking powder, and salt into the wet mixture. Mix until just combined and no dry ingredients are visible.
Pour the batter into the prepared pan and bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Remove the pudding from the oven and let it cool in the pan for 10 minutes. Then, lift it out of the pan and allow it to cool until just warm.
Cut into 9 squares and serve with vanilla ice cream, butterscotch sauce, and a sprinkle of flaky sea salt, if desired.