Heat oven to 425 F with rack in the center. Use a 12×16-inch sheet of parchment paper to line a 9-inch springform pan, centering as best you can. Press the paper into the bottom edge of the pan and flatten the creases around the side. Place the pan on a shallow baking sheet.

In a mini blender or food processor, combine 1 cup of the grapes, 1/4 cup of the sugar, cornstarch, and salt. Blend on high until smooth, about 1 minute.

In the bowl of an electric mixer fitted with a paddle, cream the cream cheese and the remaining 2/3 cup sugar until fluffy, 1 to 2 minutes. With the mixer at medium speed, gradually pour in the heavy cream; scrape down and beat until the mixture resembles soft whipped cream, about 1 minute at high speed. Beat in the eggs, one at a time, scraping down in between additions. At medium speed, gradually add the grape puree mixture; scrape and mix until well blended, 1 minute more.

Pour the batter into the prepared pan, transfer to the oven and bake 38 to 40 minutes, until cake is nicely puffed and top is a caramel-brown. Cracks will start to appear around the edge and the cake will be quite wobbly when touch gently with a finger. Set the cake pan on its baking sheet on a rack to cool. The cake will sink to half its height. Cool to room temperature, about 1 to 2 hours, and then refrigerate uncovered. Do not unmold the cake from the pan until after it is well chilled, at least 5 hours or overnight.

Slice remaining grapes in half. If desired, in a small bowl, combine grapes with orange liqueur and set aside.

When ready to serve, remove the side of the pan. Carefully pull the creased parchment away from the side of the cheesecake, then transfer cheesecake to a serving platter by peeling the parchment away from the bottom of the cake and underneath the pan while gently tugging the cake toward the platter and off the parchment.

Garnish the top of the cheesecake with grape-liqueur mixture, or fresh grapes if desired. Cut the cake into wedges with a thin, sharp knife.