Preparation
Make the ice cream
In a large mixing bowl, beat the egg yolks lightly, and set aside.
In a large saucepan, bring the milk, heavy cream, sugar, and salt to a simmer, stirring constantly, just until the sugar is dissolved, about 3 to 5 minutes. Break in the goat cheese, and whisk until melted, 2 to 3 minutes more. Remove from the heat, and while continuously whisking, very slowly ladle about a cup of the hot cream into the egg yolks. Continuing to whisk, add the remaining cream mixture to the egg yolks, then strain the liquid through a mesh strainer back into the same pot. Continue to cook so the mixture thickens slightly, about another 5 minutes. Then transfer to a heatproof bowl with lid, and chill at least 4 hours or overnight.
In a blender or food processor, add the guava paste and 1/2 cup water, and blend until smooth. Transfer puree to a small saucepan, and cook, stirring constantly, until the consistency is smooth, thick, pourable jelly, 3 to 5 minutes. Let cool completely.
Remove the goat cheese mixture from the fridge, and churn in an ice cream maker according to the manufacturer’s instructions, about 20 minutes. In a freezer container, alternate adding 1 cup ice cream with a few drizzles guava mixture. Cover and freeze at least 4 hours or overnight.
Make the shortbread
In a large mixing bowl, beat the butter, confectioner’s sugar, lemon zest, vanilla, and salt until smooth. Add the flour and cornmeal, and beat until combined. Using your hands, gather the dough, kneading lightly, and flatten into a 3×5-inch rectangle. Chill at least 30 minutes.
Place the dough between 2 sheets of parchment paper, and roll to a 13×9-inch rectangle. Remove the top sheet of parchment. Cut the dough in half lengthwise, then crosswise 2 times to make 6 squares. Cut each square in half diagonally to make 12 triangles. Transfer the cut dough on the parchment to a cookie sheet, and freeze at least 20 minutes.
Preheat the oven to 350°F. Remove the dough from the freezer, and using a spatula, carefully place the cookies on the cookie sheet. Prick the tops with a fork, and bake until golden brown around the edges, 15 to 20 minutes. Remove from the oven, transfer to a wire rack, and let cool completely.
To serve, scoop some of the ice cream between 2 cookies and press down to sandwich.
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