Start by melting your butter completely, then set it aside to cool for about 10 minutes. This step matters more than people realize, since warm butter can make your eggs scramble slightly and throw off the texture. I usually melt mine in the microwave in short bursts so it doesn't get too hot.
In your large bowl, stir the melted butter together with both sugars until it looks smooth and glossy. Add the pumpkin puree and mix well, then stir in the egg yolk and vanilla extract. I like using a spatula here instead of a whisk since the batter is fairly thick already.
In a separate bowl, whisk together the flour, pumpkin pie spice, baking soda, cream of tartar, and salt. Whisking the dry mix on its own helps distribute the leavening and spices evenly, so you don't end up with a bite that tastes too spiced or too bland compared to the rest.
Gradually fold the dry ingredients into the wet mixture, mixing just until no flour streaks remain. I try not to overmix at this stage, since that's usually what makes cookies turn out tough instead of tender. The dough will look soft and slightly sticky, which is exactly what you want.
Cover the bowl with plastic wrap and refrigerate the dough for at least an hour. This chilling step firms up the butter again so your cookies hold their shape instead of spreading into thin, flat puddles in the oven. I know it's tempting to skip, but don't.
Once chilled, preheat your oven to 350°F and line a baking sheet with parchment paper. Scoop the dough into balls, then roll each one in the cinnamon-sugar mixture until fully coated. This coating is what gives snickerdoodles their signature crackled, sparkly look.
Place the coated dough balls a couple of inches apart on your baking sheet and bake for about 10-11 minutes. The centers will look slightly underdone when you pull them out, but that's exactly what keeps them soft and chewy once they cool on the pan for a few minutes