In a heavy-bottomed saucepan, stir together the cranberries, sugar, lemon zest, lemon juice, and water. Bring the mixture to a rolling boil over medium-high heat.

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

Preheat your oven to 350°F (175°C).

In a medium mixing bowl, whisk together the flour, sugar, baking powder, and salt. Cut in the cold butter using a pastry blender or two forks until the mixture forms coarse crumbs that clump together when squeezed. Measure out 1/2 cup of the crumb mixture and set it aside for the topping.

Press the remaining crumb mixture firmly 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 remove and let cool slightly on a wire rack.

In a large mixing bowl, beat the softened cream cheese and sugar with an electric mixer on medium speed until light and airy. Beat in the egg and vanilla just until smooth. Pour the filling over the pre-baked crust and spread it evenly.

Using a teaspoon, drop spoonfuls of the cooled cranberry mixture over the cheesecake layer, leaving about a 1/2-inch border around the edge.

Sprinkle the reserved crumb topping evenly over the cranberries.

Bake for 30 to 35 minutes, until the edges are golden brown and the center is set and no longer jiggles when the pan is gently shaken.

Let the pie cool completely on a wire rack, then cover and refrigerate for at least 2 hours before slicing. Keep any leftovers refrigerated.