Preheat oven to 375 degrees. Cut two 12-by-16-inch rectangles from parchment paper. On each piece of parchment, draw 12 circles (3 1/2 inches) with a pencil. Place each piece of parchment, pencil-side down, on a baking sheet.
In the bowl of an electric mixer fitted with the paddle attachment, cream butter and sugar on medium-high speed until pale yellow, about 4 minutes. Add vanilla and egg whites; beat until combined, scraping down the sides of the bowl with a rubber spatula. Add flour, salt, ground ginger, and crystallized ginger. Beat on low speed until just incorporated.
Spoon a heaping tablespoon of batter onto each circle. With back of a spoon, spread evenly to edges.
Bake until cookies are golden brown around edges, about 12 minutes; rotate sheets twice during baking for even browning. Transfer sheets to wire racks to cool completely.
Spoon about 1/2 cup of peach ice cream on the bottom of a cookie. Top with another one, and press down gently, forcing ice cream to edges. Wrap in plastic, and place in freezer. Repeat with remaining cookies and ice cream. Freeze sandwiches at least 2 hours before serving.
These sandwiches are best eaten within two days. Let them soften in the refrigerator for 15 minutes before serving.