Brown the butter for the dough: Add the 1/2 cup butter to a light-colored saucepan over medium heat. Cook, stirring, until the milk solids turn amber and it smells nutty, 5–7 minutes. Immediately scrape into a heatproof bowl and chill in the fridge 10–15 minutes until opaque but still soft.
Prepare pans and oven: Line two baking sheets with parchment. Preheat the oven to 365°F (185°C).
Dry the pumpkin: Spread 2/3 cup pumpkin puree on a plate and blot with paper towels several times until reduced to about 1/3 cup and no visible moisture transfers easily. Set aside.
Mix the streusel: In a bowl, stir together flour, brown sugar, pumpkin pie spice, salt, and pecans. Drizzle in the melted butter and toss with a fork until clumpy. Squeeze a few big clumps with your hands. Freeze for 10 minutes to firm up.
Combine dry ingredients for the dough: In a separate bowl, whisk the flour, cornstarch, pumpkin pie spice, baking powder, baking soda, and salt until evenly blended.
Make the dough: In a large bowl, whisk the cooled browned butter with brown sugar and granulated sugar until glossy and slightly thick, about 45 seconds. Whisk in the egg yolk and vanilla until pale and smooth. Stir in the dried pumpkin until fully incorporated.
Finish the dough: Add the dry mixture to the wet ingredients and fold with a spatula just until no dry pockets remain. Let the dough rest 5 minutes to hydrate; if still very sticky, chill 10 minutes.
Shape and crown: Scoop 2-tablespoon portions (about 35–40 g), roll into balls, then flatten into 1/2-inch-thick pucks. Press the top of each puck firmly into the chilled streusel to coat heavily. Place crumb-side up on the prepared sheets, spacing about 2 inches apart. Gently press again to anchor the crumbs.
Chill the trays: Refrigerate the loaded baking sheets for 15 minutes; this keeps the cookies thick and the crumbs craggy.
Bake: Bake one sheet at a time for 10–12 minutes, rotating once, until the edges look set and the centers are just matte. Cool on the pan for 8 minutes, then transfer to a rack to cool completely.
Glaze and serve: Stir powdered sugar with 1 tablespoon milk, adding drops more until thick and drizzleable. Flick or pipe the glaze over cooled cookies and let set 10 minutes.
Tip: If your pumpkin brand is especially wet, weigh it after blotting—aim for 50–60 g (about 1/3 cup). A drier puree prevents cakey, underbaked centers.