In a medium saucepan over medium heat, combine cranberries, 3/4 cup of sugar, lemon zest, lemon juice, and water. Stir and bring to a gentle boil, then reduce heat and simmer for 10-12 minutes. Cool completely.

Preheat oven to 350°F (175°C). In a large mixing bowl, whisk together flour, 1/4 cup sugar, baking powder, and salt. Cut in butter until mixture resembles coarse crumbs.

Set aside 1/2 cup of the crumb mixture for topping. Press remaining crumbs into a 9-inch pie pan evenly. Bake for 10 minutes until lightly golden. Cool slightly.

In a large bowl, beat cream cheese with 1/2 cup sugar until smooth. Add egg and vanilla extract, mixing until well combined.

Pour cheesecake filling into the cooled crust, smoothing out evenly. Spoon cooled cranberry filling over cheesecake in dollops, then sprinkle reserved crumbs on top.

Bake assembled pie for 30-35 minutes until edges are golden and filling is set but slightly jiggly in the center. Cool at room temperature for at least 2 hours before refrigerating.