Preheat oven to 350°F and adjust oven rack to middle position. Grease and flour an 8x8-inch square or round baking pan. Line a large, deep oven pan with a clean dishtowel then place the baking dish on top of the towel.
Bring a large, covered saucepan of water to boil while preparing the cake batter.
Place half the dates in a medium bowl and cover with ¾ cup warm water. Add baking soda, stir, cover the dates and soak them for 5 minutes. Drain dates over another bowl (to reserve the water).
1¼ cups chopped pitted dates, ¾ cup warm water, ½ teaspoon baking soda
In another medium bowl, whisk together flour, baking powder and salt.
1¼ cups all-purpose flour, ½ teaspoon baking powder, ½ teaspoon kosher salt
Using a food processor fitted with the metal blade, process the dates and brown sugar until just blended (10 1-second pulses). Add reserved soaking liquid, eggs, and vanilla and process 5 seconds or until smooth. While the food processor is running, pour the melted butter in and process for 3 seconds. Transfer this mixture to the softened dates.
¾ cup dark brown sugar, 2 large eggs, 1½ teaspoons pure vanilla extract, 4 tablespoons unsalted butter
Add the dry ingredients to wet and mix just until mixtures blend.
Pour the batter into the prepared baking dish and very carefully pour the boiling water halfway up the sides of the baking dish. Do not get any of the boiling water in the batter. Cover the large pan, which includes the baking dish, tightly with aluminum foil.
Bake 40-45 minutes or until the outer 2 inches of the cake develops small holes and the center is firm to the touch.
While cake is baking, make the toffee sauce and keep it warm until cake is baked.
In a medium saucepan over low-medium heat, melt butter. Whisk in brown sugar and whisk until butter and brown sugar fully incorporated. Cook, whisking occasionally, until mixture looks puffy and sugar is dissolved, about 3-4 minutes.
8 tablespoons unsalted butter, 1 cup dark brown sugar
Remove from heat and slowly, while quickly whisking, add cream and rum. Continue to whisk until mixture comes together.
⅔ cup heavy cream, 1 tablespoon rum
Place pan on stovetop, over low heat, and simmer until frothy, about 3 minutes. Whisk occasionally.
Remove from heat and stir in lemon juice. Cover pan with paper towel and with the lid (be sure heat is off). Set sauce aside until ready to use.
½ teaspoon fresh lemon juice
Once the cake has baked, immediately (and very carefully) remove the baking dish from the water bath and set on wire rack. Poke several small holes through the top of the cake. Pour all except ⅓ cup warm sauce evenly over the toffee pudding.
Cool for 10 minutes, cut, and serve with ice cream and drizzle with extra sauce.
1 pint vanilla ice cream