In a medium saucepan placed over medium heat, add the half-and-half and 1 ½ cups of heavy cream. Add the sugar and whisk until dissolved. Add the cinnamon sticks to steep.
While the cream heats, add 2 egg yolks to the remaining ½ cup heavy cream., whisking to combine. When the cream is hot, slowly ladle some of the hot cream into the egg yolk mixture, whisking continuously. This will temper the eggs.
Then, slowly pour the egg yolk mixture back into the hot cream while whisking. Cook over medium-low heat until slightly thickened, being careful not to boil the cream mixture.
Remove the pan from the heat and cool for 10-15 minutes (the cinnamon sticks will continue to steep while the mixture cools). Strain the mixture into a heat-proof bowl. Stir in the ground cinnamon and vanilla extract.
Cover with plastic wrap, pressing the plastic down directly on top of the cream mixture. Refrigerate until completely chilled.
Transfer the mixture to an ice cream machine. Churn until the mixture is the consistency of soft serve. Transfer to a quarter sheet pan and spread into an even layer. Freeze until firm, about 4 hours, or overnight.
In a mixing bowl, combine the flour, salt, baking soda, baking powder, cinnamon, cloves, and ginger. Whisk to mix thoroughly. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, combine the butter and sugar. Cream together until well blended. Add the egg, molasses, and vanilla extract and mix until incorporated.
Slowly add in the flour mixture and mix until just combined. Chill for 1 hour.
Remove cookie dough from refrigerator 20 minutes before baking. Preheat oven to 350 degrees.
Using a tablespoon-sized cookie scoop, portion the dough into balls. Roll the balls in the coarse sugar and place onto parchment-lined baking sheets. Bake for 11-12 minutes until lightly brown around the edges. Remove and let sit for 3-4 minutes. Transfer the cookies to a cooling rack to cool completely.
This recipe yields approx. 28 cookies, 16 of which will be used to make the ice cream sandwiches. Store the remaining cookies in an airtight container to enjoy separately.
Assembling ice cream sandwiches
Remove sheet pan of ice cream from the freezer. Using a 3-inch biscuit cutter (to match the size of the baked cookies), cut disks of ice cream. (You will get 8 3-inch disks of ice cream from the quarter sheet pan. Save the remaining ice cream to enjoy separately.
Using a spatula, transfer the disks of ice cream to a cookie, placed bottom-side up. Top with another cookie. Freeze the ice cream sandwiches until firm.
Prep and cook times do not include the chill and freeze times for the ice cream, nor the chill time for the cookie dough.