In a small, shallow bowl, whisk together the boiling water or brewed coffee, espresso powder, and apple juice. Set aside to cool slightly.
Place the room-temperature mascarpone in the bowl of a stand mixer fitted with the whisk attachment. Gently stir with a flexible spatula first to loosen the mascarpone.
Add the brown sugar, cinnamon, ginger, nutmeg, cloves, and salt. Whisk on medium speed until evenly combined.
Add the vanilla extract. Pour in the heavy cream in two additions, whisking between each addition until incorporated.
Add the pumpkin puree and continue whisking on medium speed until the pumpkin mascarpone mixture reaches soft to medium peaks. Avoid overwhipping.
Dip half of the ladyfingers individually into the coffee mixture for about 2–3 seconds each. Arrange them snugly across the bottom of a 2-quart rectangular baking dish, breaking individual ladyfingers when necessary.
Spread half of the pumpkin mascarpone cream evenly over the soaked ladyfingers.
Repeat with another layer of dipped ladyfingers, followed by the remaining mascarpone mixture. Smooth the surface with a spatula.
Cover tightly and refrigerate the Pumpkin Tiramisu for at least 4 hours, although chilling it overnight produces an even softer texture and more developed flavor.
Just before serving, generously dust the surface with Dutch-process cocoa powder using a fine-mesh sieve.