For the Sweet Summer Corn Ice Cream:
3 large ears sweet corn, husked
3 1/2 cups whole milk
1 cinnamon stick
6 large egg yolks
1 1/2 cups sugar, divided
1 teaspoon sea salt
1 cup heavy whipping cream or half and half
For the Salted Butter Cookies:
10 tablespoons European-style salted butter (such as Lurpak), softened
2/3 cup sugar
1/4 teaspoon sea salt
1 large egg
2 cups flour
Fleur de sel or coarse sea salt for sprinkling (optional)
Prepare the ice cream: Using a serrated knife, slice the kernels off of the corn cobs into a large saucepan. Break the corn cobs in half, and add them to the saucepan. Pour the milk over the corn, add the cinnamon stick, and set the saucepan over medium heat. Bring the mixture to a simmer, stirring occasionally, cover the saucepan, and turn off the heat. Allow the mixture to steep for 1 hour.
Remove the corn cobs and cinnamon stick from the mixture and discard. Using either an immersion blender or a traditional blender, pulse the milk and corn kernels until the corn is broken up into small pieces.
Set a mesh strainer over a large bowl, and pour the mixture over the strainer, pressing gently on the solids to realease as much of the liquid as possible. Disard the solids.
In a medium bowl, whisk together the egg yolks, 1/2 cup of sugar, and the salt. In a large saucepan, combine the milk mixture, the cream, and the remaining sugar. Bring the mixture to a simmer over medium heat, stirring to dissolve the sugar.
Temper the egg yolks by gradually pouring half of the hot milk mixture into the bowl, whisking constantly. Gradually return this mixture to the saucepan and cook over medium heat, whisking constantly, until the mixture thickens (do not boil!), 3-4 minutes.
Transfer the mixture to a heat-proof bowl and allow to cool for 10 minutes. Press plastic wrap onto the surface of the custard and chill until very cold, at least 6 hours.
Freeze the ice cream in your ice cream maker according to the manufacturer's instructions. Transfer to a freezer-safe container and freeze until firm enough to scoop and serve.
Prepare the cookies: In the bowl of an electric mixer fitted with the paddle attachment, beat the butter and the sugar until light and fluffy, about 3 minutes. Add the salt and the egg, and beat until smooth, stopping to scrape the sides of the bowl as necessary. With the mixer on low speed, mix in the flour until combined.
Divide the dough in half, and shape each half into a flat disk. Wrap each disk in plastic wrap and refrigerate for at least 1 hour.
Preheat the oven to 350F degrees. Line two baking sheets with parchment paper, and lightly flour a work surface. Roll one of the disks of dough out to a 1/4-inch thickness. Using a round, fluted, or shaped small (1-2 inches) cookie cutter, cut out shapes and transfer them to the prepared baking sheets. If desired, brush the tops of the cookies lightly with water and top each cookie with a few grains of sea salt or Fleur de Sel. Repeat the process with the remaining dough, re-rolling scraps as necessary. Bake the cookies for 7-9 minutes, until set and very lightly golden. Allow to cool, and serve with the Sweet Summer Corn Ice Cream.