Position a rack in the center of the oven and heat the oven to 300°F. Line 12 standard muffin tins (approximately 23/4 inches in diameter) with foil liners and coat lightly with cooking spray.
In the bowl of an electric stand mixer fitted with the paddle attachment (or in a large mixing bowl using a hand-held electric mixer) beat the cream cheese on medium-high speed until very smooth and fluffy, stopping to scrape down the bowl as necessary, about 4 minutes. Add the sugar, vanilla, and salt, and continue beating until well blended and smooth, scraping down the sides of the bowl frequently, about 1 minute; there should be no lumps. Add the eggs, one at a time, beating on medium speed until just blended. (Don’t overbeat once the eggs are added or the cheesecakes will puff and crack during baking.)
Transfer 2/3 cup of the batter to a small bowl. Add the pumpkin, flour, cinnamon, ginger, and nutmeg to the small bowl and stir with a wooden spoon until well blended.
Divide the plain batter among the muffin cups (about 2 generous Tbs. in each). Then divide the pumpkin batter evenly among the cups (about 1 generous Tbs. in each). Drag the tip of a wooden skewer, toothpick, or paring knife through the two batters in a random, swirly pattern to create a marbled look
Bake until the centers of the cheesecakes barely jiggle when nudged, 15 to 18 minutes. Set the muffin tins on a rack and let cool completely. Cover and refrigerate until very cold, at least 6 hours or up to 3 days.
The baked cheesecakes can be refrigerated, covered for 3 days or frozen for 1 month. Freeze the cooled cheesecakes in the tins in heavy-duty zip-top plastic bags, or remove them from the tins and arrange in airtight containers.