To Make the Ice Cream

Combine the milk, cream, lemon zest, and ½ cup of the sugar in a heavy saucepan. Cook, stirring frequently, over medium heat until almost simmering. In a bowl, whisk together the egg yolks, the remaining ½ cup sugar, and the salt. Slowly pour the hot liquid into the egg mixture, whisking as you pour. Return the mixture to the saucepan. Cook, over medium-low heat, stirring constantly with a heat-resistant plastic or a wooden spatula, until the custard reaches 175°F and lightly coats the spatula.

Strain the custard into a clean bowl and cool over an ice bath until room temperature. Stir in the lemon juice. Refrigerate the custard for at least 4 hours or up to overnight. Churn in an ice cream machine according to the manufacturer’s instructions. Freeze until scoopable, about 4 hours, depending on your freezer.

To Make the Cookies

Sift together the flour, baking soda, cinnamon, ginger, white pepper, and allspice. Stir in the salt.

Beat the butter, ¼ cup of the granulated sugar, and the brown sugar together until smooth. Stir in the egg. Mix in the molasses. Stir in the dry ingredients in 2 additions. Refrigerate the dough until very firm, at least 3 hours.

Divide the dough in half, wrap each half in plastic wrap, and refrigerate the dough for at least 2 hours or up to 4 days. When firm, roll the dough on a lightly sugared work surface into two 9-inch-long logs.

Preheat the oven to 350°F. Line 3 baking pans with parchment paper. Cut the dough into ½-inch-thick slices. There should be at least 24 cookies. Put the remaining ¼ cup granulated sugar in a bowl. Coat the cookies with the sugar. Place the cookies 2½ inches apart on the prepared pans. Bake the cookies until set and no longer wet looking, 10 to 12 minutes. They will be puffy when you take them out of the oven and will sink as they cool.

In Advance

The cookie dough can be rolled up to a week in advance and refrigerated. The sandwiches can be assembled a couple of days in advance.