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1
Preheat oven to 350 degrees. Line 2 baking sheets with parchment paper. Set aside.
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2
In a small bowl, whisk together flour, cornmeal, basil, lemon zest, salt, and ginger.
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3
In a large bowl, beat butter and 3/4 cup sugar at medium-high speed mixture is light and fluffy, approximately 4 minutes. Add egg, beating until well combined. Reduce speed to low, and slowly add flour mixture, beating until well combined.
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4
Roll dough into 1-1/2-inch balls, and roll in remaining 1/2 cup sugar. Place 1-1/2-inches apart on prepared baking sheets. Using a flat-bottomed glass sprayed with nonstick cooking spray, slightly flatten each dough ball.
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5
Bake until edges are golden brown, 18 to 24 minutes. Transfer to a wire rack to cool completely.
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6
In a medium heavy-bottomed saucepan, combine half-and-half and 3/4 cup sugar. Heat mixture over medium-high heat until it just begins to boil; remove from heat.
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7
In a medium bowl, whisk egg yolks and remaining 1/4 cup sugar until light and fluffy. Using a ladle, pour 1 cup hot half-and-half mixture in a slow steady stream into yolks, whisking constantly. Add another cup of hot half-and-half mixture, whisking to combine. Return mixture to saucepan.
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8
Cook over medium heat, stirring constantly, until thick enough to coat the back of a wooden spoon or temperature reaches 170 degrees on an instant read thermometer.
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9
Pour mixture through a mesh sieve, and strain into a stainless-steel bowl set in an ice-water bath. Stir in peaches. Let cool completely, stirring often. Refrigerate for 2 hours before freezing.
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10
Freeze mixture in batches in a 1-quart gel-type ice-cream maker, according to manufacturer's instructions. If a firmer texture is desired, transfer ice cream to an airtight container, and freeze for 2 hours or until firm.
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11
Place a scoop of Old-Fashioned Peach Ice Cream on flat side of 1 cookie. Top with flat side of another cookie. Repeat with remaing cookies and ice cream. Serve immediately.
