These cakey cookies partner well with citrus and other fruit flavors, as shown here, but also with coffee, green tea, or nearly any ice cream you like.
Preheat oven to 350 degrees. Butter a 10-by-15-inch rimmed baking sheet and line with parchment; butter parchment. In a bowl, whisk together flour, cinnamon, ginger, and salt. In another bowl, whisk together butter, sugar, molasses, lemon zest and juice, and eggs. Whisk dry ingredients into wet ingredients until smooth. Spread batter evenly in prepared sheet.
Bake until cake is dry to the touch and edges begin to pull away from sheet, 12 to 14 minutes. Let cool 5 minutes. Loosen edges with a knife; flip out of sheet. Remove parchment; flip top-side up onto a wire rack. Let cool completely.
Transfer cake to a work surface; cut in half crosswise. Place one half top-side down on a piece of plastic wrap. Beat ice cream until malleable, then spread over cake in a 1-inch-thick layer (go all the way to edges; don't worry if a bit oozes out, as it will be trimmed later). Top with second half of cake, top-side up. Wrap in plastic and freeze at least 8 hours and up to 1 week. Use a serrated knife to trim edges and cut into 1 1/4-by-3-inch pieces. Serve immediately, or wrap in plastic and freeze for up to 1 week.
Make your own variation by swirling together 1 pint softened vanilla ice cream and 1 pint mandarin-orange sorbet. Or soften 1 pint vanilla ice cream and swirl in 1/3 cup high-quality jam.