Heat the oven to 350°F (175°C). Grease and line a 9-by-13-inch baking pan with parchment.

Place the chopped dates in a bowl, pour over the boiling water, stir in the baking soda, and let stand for 10 minutes. Mash or briefly blend until mostly smooth.

Whisk the flour, baking powder, cinnamon, nutmeg, ginger, and salt together in a medium bowl.

Cream the softened butter with both sugars until light. Beat in the eggs one at a time, then mix in the pumpkin purée and vanilla.

Add the dry ingredients in two additions, alternating with the date mixture. Mix only until combined.

Spread the batter in the prepared pan and bake for 35–40 minutes, until the center springs back and a tester comes out with a few moist crumbs.

Melt the sauce butter with the brown sugar, heavy cream, and salt. Simmer gently for 2–3 minutes, stirring, then add the vanilla.

Poke the warm cake all over with a skewer or fork. Pour about half of the warm toffee sauce over the cake and let it absorb; reserve the rest for serving.

Brown the frosting butter in a saucepan until amber and nutty-smelling. Pour it into a bowl and cool until solid but still soft.

Beat the browned butter with the cream cheese, vanilla, and salt. Gradually add the confectioners’ sugar, adding cream or milk only if needed for a spreadable consistency.

Cool the cake completely before frosting. Drizzle with additional warm toffee sauce just before serving.