Preheat oven to 375°F (190°C). Line your muffin pan with paper baking cups and set to the side.
Add the flour, baking soda, baking powder, pumpkin pie spice, and salt in a medium bowl and whisk together. Set to the side.
Add the pumpkin puree, sourdough discard, brown sugar, and granulated sugar in a large bowl, and whisk together until fully incorporated. Whisk in eggs, vanilla extract, and oil. Slowly add in the flour mixture, whisking until there are no lumps, making sure not to over-whisk the batter. Fill muffin tins about three-quarters full.
Take a medium bowl and beat the cream cheese until smooth. Add the sugar, vanilla extract, and egg yolk, then beat until thoroughly mixed (If you don't have a hand or stand mixer, warm the cream cheese in the microwave slightly and beat with a whisk).
Top each muffin with a dollop (or 1-2 tablespoons) of the cream cheese mixture, swirling it into the batter with a toothpick or knife tip. Don't worry if they don't look perfect—they will bake up nicely.
Bake muffins for 18 minutes. Check readiness by inserting a toothpick into the center; they're done if it comes out clean or has a few moist crumbs. Otherwise, bake an additional 2 minutes.