Preheat your oven to 350°F (175°C). Grease a 9-inch square baking pan with baking spray and line with parchment paper.

Finely grind the unsalted pistachios until they resemble a fine crumb using a food processor.

Whisk together all-purpose flour, baking powder, and salt in a separate bowl until well combined.

Cream together softened unsalted butter, vegetable oil, granulated sugar, and lemon zest on medium-high speed until light and fluffy.

Add the large eggs one at a time, mixing thoroughly after each addition. Then add sour cream and pure vanilla extract.

Gradually add the dry mixture to the wet mixture, alternating with buttermilk. Mix on low speed until just combined.

Pour the cake batter into the prepared baking pan, spreading it out evenly.

In a medium bowl, beat the cream cheese until smooth, then add the large egg and vanilla extract, mixing until combined.

Gradually add the powdered sugar, beating until thick and creamy.

Pour the cream cheese mixture over the cake batter and smooth it out.

Bake for 40-45 minutes, checking for doneness with the top lightly cracked and the center slightly jiggly.

Let the cake cool in the pan for 10-15 minutes before transferring to the refrigerator to chill for at least 2 hours.

Remove from the refrigerator, slice into squares, and serve chilled or at room temperature.