1. Preheat the oven to 400°F (200°C). Grease a 9-inch springform pan and line the bottom with parchment paper.

2. In a large bowl, beat the cream cheese with a hand mixer until smooth and creamy. Gradually add in the 1 cup of sugar and cornstarch, and mix until well combined.

3. Add the vanilla extract and eggs one at a time, beating well after each addition.

4. In a small saucepan, combine the heavy cream and Earl Grey tea leaves. Heat over medium, stirring occasionally, until the mixture just comes to a simmer. Remove from heat and let steep for 15 minutes.

5. Strain the tea-infused cream through a fine-mesh sieve, pressing on the solids to extract as much liquid as possible. Discard the tea leaves.

6. Slowly pour the tea-infused cream into the cream cheese mixture, beating until fully incorporated.

7. Pour the batter into the prepared springform pan and bake for 50-60 minutes, until the edges are set but the center is still slightly jiggly.

8. Turn off the oven and leave the cheesecake inside with the door ajar for 1 hour.

9. Remove the cheesecake from the oven and let it cool completely on a wire rack, then refrigerate for at least 4 hours or overnight.

10. In a small saucepan, combine the blackberries, 1/4 cup sugar, water, and lemon juice. Cook over medium heat, stirring occasionally, until the berries have broken down and the sauce has thickened, about 10-15 minutes.

11. Remove the cheesecake from the springform pan and serve slices drizzled with the warm blackberry sauce.