Butter eight 3-by-2-inch ramekins; sprinkle 1 teaspoon sugar in the bottom of each; set aside. Place kumquats, remaining cup sugar, and 1/2 cup water in a saucepan over medium heat. Bring to a boil; reduce heat to low. Simmer until kumquats are soft, 15 to 20 minutes. Place kumquats in bottom of each ramekin; set aside. Reserve liquid for poaching whole kumquats for garnish, if desired.
Heat oven to 350 degrees. Combine dates and baking soda in a bowl; add 1 cup boiling water; stir to combine; set aside. Sift flour, baking powder, and salt in a bowl; set aside. In an electric mixer fitted with the paddle attachment, beat 4 tablespoons butter and brown sugar until fluffy. Add eggs one at a time, beating after each addition. Add flour mixture; beat until combined. Add date mixture and walnuts; beat until combined. Pour batter into ramekins. Bake until a cake tester inserted in the middle comes out clean, about 30 minutes. Run a knife around puddings' edges; unmold. Drizzle with toffee sauce.