Inspired by the cardamom-date cakes found in the Arabian Gulf, this dessert derives its intoxicating flavor from Medjool dates, Arabic coffee, and semisweet chocolate.
Make the chai ice cream, if desired: Combine milk, tea, cardamom, cinnamon, and ginger in a medium saucepan. Split vanilla bean lengthwise and scrape seeds into the mixture using the back of a knife, then add pod to mixture and bring to a boil. Cover and let steep at least 30 minutes and up to 1 hour.
Prepare an ice bath. In a medium bowl, whisk together egg yolks and sugar until light and fluffy. While whisking, slowly add a little of the warm milk mixture into the yolk mixture to temper. Continue this process until all of the milk mixture has been added. Pour mixture back into pan and cook over medium heat, stirring constantly until mixture is thick enough to coat the back of a spoon, 3 to 5 minutes. The custard should retain a line drawn across the back of the spoon with your finger. Strain mixture through a fine-mesh sieve into a medium bowl set in the ice bath; discard solids. Add cream and stir occasionally until cooled. Cover and refrigerate until chilled, at least 1 hour and up to overnight.
Pour into an ice cream maker and process according to the manufacturer's instructions until set but not hard. Transfer soft ice cream to a container and freeze at least 4 hours and up to overnight.
Make the cake: Preheat oven to 350 degrees. Butter a 9-cup Bundt pan and dust with cocoa powder and flour, tapping out excess. In a large glass measuring cup or medium bowl, combine dates, hot water, coffee, and cocoa and let stand 10 minutes. In another medium bowl, whisk together flour, salt, cinnamon, and cardamom. Using an immersion blender, puree the date mixture until smooth. Add baking soda and stir to combine.
In the bowl of an electric mixer fitted with the paddle attachment, cream together butter and dark-brown sugar until light and fluffy, about 2 minutes. Add the eggs, one at a time, scraping down the sides of the bowl after each addition. Beat well to combine (batter will look curdled). Add the flour mixture in two additions, alternating with the date mixture, beginning and ending with the flour. Beat in the chocolate.
Transfer the batter to prepared pan. Bake until a cake tester inserted in center comes out with moist crumbs attached, about 30 minutes.
Meanwhile, make the glaze: Bring butter, dark-brown sugar, date syrup, salt, and 1/4 cup water to a boil in a saucepan over medium-low heat, stirring occasionally. Boil 3 minutes. Remove from heat. Whisk in cream. Add chocolate and whisk until smooth.
Transfer cake pan to a wire rack to cool for 10 minutes. Invert cake onto rack, and let cool 20 minutes (the cake should still be warm). Poke cake all over with a skewer or toothpick. Pour half of the glaze into the bottom of the pan and return cake to pan. Pour remaining glaze over top and allow to soak into cake, about 20 minutes. Invert cake onto plate or cake stand and serve warm or at room temperature with chai ice cream, if desired.