Preheat the oven to 350℉. Line a 12-cup muffin pan(s) with parchment paper muffin liners. Set aside.
In a large bowl, add the pure pumpkin, sugar, oil, and large eggs. Use an electric mixer to beat until thoroughly combined, fluffy looking, and lighter in color; about 1-2 minutes.
1 can (15 oz) pure pumpkin, 1¾ cups granulated sugar, 1 cup vegetable oil, 4 large eggs
In a separate small bowl, whisk together the all-purpose flour, baking powder, baking soda, cinnamon, pumpkin pie spice, and salt.
2 cups all-purpose flour, 2 teaspoons baking powder, 1 teaspoon baking soda, 1 teaspoon ground cinnamon, 1 teaspoon pumpkin pie spice, 1 teaspoon salt
Add half of the dry ingredients into the larger bowl of wet ingredients and beat on low speed just until a few flour streaks remain in the batter. Add the remaining dry ingredients and beat on low speed just until combined and no flour streaks or pockets remain in the batter.
Make the cheesecake swirl by adding the cream cheese, powdered sugar, large egg, and vanilla extract into a mixing bowl, and beat until creamy and combined. *I re-use the bowl that had the dry ingredients in it to make the cheesecake swirl mixture. 1 bar (8 oz) full-fat cream cheese, 1 cup powdered sugar, 1 large egg, ¼ teaspoon vanilla extract
Fill each muffin cup halfway with the pumpkin mixture. Add 1 tablespoon of the cheesecake mixture to each muffin cup. Add more pumpkin muffin batter on top of the cream cheese in each muffin cup to completely cover it. Each muffin cup should be almost full.
Use a toothpick to swirl the two batters together. Do not over swirl, just swirl the batters until you start to see the swirl lines of them combining.
Bake for 18-22 minutes. Use a toothpick to ensure the muffins are cooked thoroughly. The cheesecake will naturally be softer, so make sure you are testing the pumpkin muffin itself, and a toothpick inserted should come out clean with moist crumbs and no wet batter.
Let the muffins cool in the muffin pan for 10 minutes, and then transfer them to a wire cooling rack to cool completely to room temperature. These muffins can be eaten slightly warm, at room temperature, or chilled. They are delicious either way!*Because of the cream cheese, the muffins should be stored in the fridge within 2 hours of being at room temperature. See notes for more details.