Preheat oven to 350°. Butter a 9-in. round baking pan and line bottom with parchment paper.
Grind tea leaves to a fine powder with a mortar and pestle or in a clean coffee grinder. In a large bowl, whisk tea with flour, baking powder, baking soda, salt, and chai masala. Peel and core apples, then cut into 1/4-in. dice. Put in a medium bowl and toss with 2 tbsp. of the flour mixture.
Using a stand mixer fitted with the paddle attachment, cream butter and brown sugar on medium-high speed until light and fluffy, 2 to 3 minutes. Add eggs one at a time, beating after each addition. Stop mixer, add flour mixture, and beat on low just until no streaks of flour are visible, about 1 minute. Remove bowl from mixer and fold in apples.
Pour batter into prepared pan and level it with an offset spatula. Bake, rotating halfway through, until cake is golden and a skewer inserted into center comes out with some crumbs adhering, about 40 minutes. Cool in pan on a wire rack 5 minutes. Run a thin knife around inside of pan to release cake, then invert onto rack, remove parchment, and cool completely.
Before serving, dust liberally with powdered sugar.
*Find chai masala at Indian grocery stores, or make your own: In a clean coffee grinder, whirl 10 green cardamom pods, 1 black cardamom pod (or use 2 whole green cardamom pods), 6 black peppercorns, 4 cloves, and a 2-in. cinnamon stick until finely ground. Add 1 tbsp. ground ginger and whirl to blend. Makes about 2 tbsp.
MAKE AHEAD Up to 3 days at room temperature, in an airtight container lined with a clean kitchen towel to absorb moisture.