Heat the oven to 350º F. Butter 6 (6 ounce) custard cups and arrange on baking sheet.
To make cakes: In a medium bowl, combine dates and hot water and let steep just to soften the dates, about 10 minutes. In a food processor fitted with the metal blade, add dates with the hot water, espresso powder and vanilla bean paste and purée, about 30 seconds.
In the bowl of a stand mixer fitted with a paddle, cream butter and sugar together until light and fluffy, about 5 minutes. Add eggs, a tablespoon at a time, beating well between each addition and scraping down the sides as needed. Add date purée and mix lightly. Stir in flour, baking powder and salt. The mixture will be quite sloppy.
Divide evenly between the 6 prepared custard cups, about 4 ounces in each cup. Each cup should be just over half full. Using the back of a spoon, make a hollow in the center using the back of a spoon to allow for even baking without a dome. Bake at 350º F for 35 minutes or until risen and golden.