Begin by making the chai soak. To a small saucepan, add the water, chai tea, cinnamon sticks, cardamom pods, and sugar. Bring to a boil, then reduce the heat and simmer for 15 minutes. Remove the pan from the heat and steep the chai soak for a further 5 minutes. Strain the soak into a shallow bowl through a fine-mesh sieve and stir in the espresso.
In the bowl of a stand mixer fixed with a whisk attachment, whip the egg yolks and 50 grams of the sugar together until light and fluffy. This will take about 5 minutes. Transfer the egg yolk mixture to a large mixing bowl, making sure to scrape the sides of the mixer bowl well.
To the stand mixer's bowl, add the whipping cream and remaining 50 grams of sugar. Whip on medium-high until peaks are just beginning to form. Add the ground cardamom, vanilla extract, and mascarpone, then continue whipping until stiff peaks form.
Fold the mascarpone whipped cream into the whipped egg yolk mixture until fully combined.
Prepare your assembly station with the ladyfingers, chai soak, chai cream, and an 8-inch square baking dish. Dust the baking dish with 1 tablespoon of cocoa powder.
Assemble by dipping the ladyfingers into the chai soak, turning to make sure both sides are covered. Carefully transfer the soaked ladyfingers to the cocoa-dusted baking dish. Repeat until the bottom of the dish is covered with chai-soaked ladyfingers. You may have to break some of the ladyfingers up to cover the entire bottom of the dish.
Cover the layer of chai-soaked ladyfingers with half of the chai cream, using an offset spatula or spoon to smooth the cream out evenly over the ladyfingers. Repeat this process starting with another layer of chai-soaked ladyfingers followed by the remaining half of the chai cream. Dust the remaining cocoa powder over the top of the tiramisu.
Cover and refrigerate the tiramisu for at least 5 and up to 24 hours before serving. Leftover tiramisu will keep in the fridge for up to three days.