In a heavy-bottomed saucepan over medium-high heat, combine the cranberries, sugar, lemon zest, lemon juice, and water. Stir occasionally until the mixture reaches a boil.

Reduce the heat to medium-low and let it simmer gently for about 10 minutes, until the cranberries soften and begin to pop. Remove from the heat and set aside to cool completely.

Heat the oven to 350°F (175°C). In a medium bowl, whisk together the flour, sugar, baking powder, and salt.

Add the cold butter to the flour mixture and work it in with a pastry blender or your fingertips until the texture is crumbly and clumps together when pressed. Measure out 1/2 cup of the crumb mixture and reserve it for the topping.

Press the remaining crumb mixture firmly and evenly into the bottom and up the sides of a 9-inch pie pan or a 9-inch tart pan with a removable bottom. Bake for 10 minutes, then set aside to cool slightly.

In a large bowl, beat the softened cream cheese and sugar with an electric mixer on medium speed until light and fluffy. Add the egg and vanilla and continue mixing until smooth and well combined. Pour the filling over the pre-baked crust and spread it into an even layer.

Using teaspoons, drop spoonfuls of the cooled cranberry mixture evenly over the cheesecake filling, leaving a 1/2-inch border around the edge.

Sprinkle the reserved crumb mixture over the top of the cranberries.

Bake for 30 to 35 minutes, until the edges are golden brown and the center is just set. Let the pie cool on a wire rack.

Cover and refrigerate for at least 2 hours before slicing and serving. Keep any leftovers chilled.