Adjust oven rack to middle position and heat oven to 350 degrees. Grease and flour eight 6-ounce ramekins. Fold dish towel in half and place in bottom of large roasting pan. Place prepared ramekins on top of towel; set aside pan. Bring kettle of water to boil.
Combine half of dates, warm water, and baking soda in 2-cup liquid measuring cup (dates should be submerged beneath water); soak dates for 5 minutes. Meanwhile, whisk flour, baking powder, and salt together in large bowl.
Process sugar and remaining dates in food processor until no large chunks remain and mixture has texture of damp, coarse sand, about 45 seconds, scraping down sides of bowl as needed. Drain soaked dates, adding the soaking liquid to processor. Add eggs, melted butter, and vanilla and process until smooth, about 15 seconds. Transfer sugar mixture to bowl with flour mixture and sprinkle soaked dates on top. Using rubber spatula, gently fold sugar mixture into flour mixture until just combined and date pieces are evenly dispersed.
Divide batter evenly among prepared ramekins (ramekins should be two-thirds full). Quickly pour enough boiling water into roasting pan to come ¼ inch up sides of ramekins. Cover pan tightly with aluminum foil, crimping edges to seal. Bake until cakes are puffed and surfaces are spongy, firm, and moist to touch, about 40 minutes. Immediately transfer ramekins from water bath to wire rack and let cool for 10 minutes.