How to Make It
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.
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.
In a medium bowl, using an electric mixer, beat the cream until almost firm. Fold the cream into the cooled custard.
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.
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.