1. In a large saucepan, steam the squash until barely tender, about 7 minutes. Let cool slightly and pat dry. In a large nonstick skillet, melt the butter with 1/4 cup of the sugar. Add the squash and cook over moderately high heat, stirring occasionally, until golden, about 12 minutes. Add the ginger and cook, stirring, for 1 minute. Transfer two-thirds of the candied squash to a food processor and puree until smooth.
2. In a medium saucepan, whisk the egg yolks with the remaining 1/2 cup plus 2 tablespoons sugar and cook over low heat, stirring constantly, until warmed through, about 1 minute. Stir the cornstarch into the milk, then whisk the mixture into the saucepan and bring to a boil over moderate heat, whisking constantly. Cook until thickened, about 7 minutes; the custard should have the consistency of thin sour cream. Whisk in the squash puree and vanilla. Strain through a fine-mesh sieve and let cool.
3. In a medium bowl, using an electric mixer, beat the cream until almost firm. Fold the cream into the cooled custard.
4. Spoon 1 1/2 tablespoons of the cookie crumbs into each of ten 6-ounce glasses. Top with 1/3 cup of the custard and gently tap each glass to form even layers. Repeat with another 1 1/2 tablespoons of the cookie crumbs and a large dollop of custard. Cover the glasses with plastic wrap and refrigerate until the crumbs soften, at least 5 hours. Reserve the cookie crumb and candied squash garnishes at room temperature.
5. Sprinkle the candied squash on top of each parfait. Garnish with the remaining cookie crumbs and serve.
Make Ahead: The parfaits and candied squash can be refrigerated separately for up to 3 days.