For the ice cream: In a bowl, stir together the cornstarch and 1/4 cup of the milk, and set aside. Cut the kernels off the corn cob and cut the cob into large chunks. In a medium pot, whisk together the remaining milk, cream, sugar, syrup, and salt. Add the corn kernels and cob, and bring to a boil over medium-high heat. Cook for 4 minutes, stirring continuously. If the mixture becomes too frothy and threatens to boil over, partially or totally remove from heat for a few seconds until it calms. I found that scooting the pot halfway off the burner and continuing to cook worked well.
After 4 minutes, add the slurry. Return to a boil and cook for about 2 minutes more, stirring continuously again, until thickened. Remove the cob, then pour the hot mixture into a large bowl. (Alternatively, strain out all corn solids by pouring the mixture through a fine mesh sieve.)
Place cream cheese in a bowl and pour in a small amount of the hot milk mixture. Whisk until smooth, or use an immersion blender to get out the lumps. Whisk in the remaining milk mixture, then pour mixture into a plastic bag, seal, and submerge in a bowl of ice water until chilled.
Once cold, pour mixture into an ice cream maker and process according to manufacturer’s instructions. After churning, freeze in a storage container while you make the cookies — at least 2-3 hours.
To assemble: Once cookies have cooled, remove ice cream from freezer and let soften slightly. Place a scoop of ice cream between two similar-sized cookies and squeeze gently to press them together. Repeat with remaining cookies and ice cream.
Place assembled ice cream sandwiches on a baking sheet and cover with plastic wrap, then replace in the freezer for 1-2 hours, or until hardened. After that, enjoy or wrap individually in plastic wrap for later.
Cookies are best in the sandwich when on the underbaked and softer side — if they’re too soft to stand up to the ice cream at room temperature, pop in the freezer for 10-15 minutes before assembling sandwiches.