How to Make It
Make the cajeta In a medium saucepan, combine the goat milk with the granulated sugar and lemon zest and bring to a simmer over moderate heat. Remove the pan from the heat and whisk in the baking soda and the vanilla bean seeds. Return the pan to moderately low heat and simmer, whisking occasionally, until the caramel is amber and the consistency of condensed milk, about 1 hour and 30 minutes. Set a fine sieve over a small heatproof bowl. Strain the caramel into the bowl; discard the solids. Let cool slightly. Cover with plastic wrap and refrigerate until firm and cold, at least 1 hour.
Meanwhile, make the cookies In a medium bowl, whisk the flour with the cornstarch, ground ginger, cinnamon, cloves and baking soda. In a stand mixer fitted with the paddle, beat the butter with the brown sugar at medium speed until light and fluffy, about 2 minutes. Add the molasses and beat until incorporated. Add the egg and vanilla paste and beat until smooth. Beat in the dry ingredients at low speed until just combined. Stir in the candied ginger. Cover the bowl with plastic wrap and refrigerate until cold, at least 1 hour.
Preheat the oven to 325°. Line 2 baking sheets with parchment paper. Spread the turbinado sugar in a shallow bowl. Roll tablespoons of dough into balls, then coat in the turbinado sugar; transfer to the prepared baking sheets, spaced 1 inch apart. Freeze for 15 minutes.
Bake the cookies, rotating the baking sheets halfway through, for about 10 minutes, until crisp on the outside but still tender on the inside. Remove the baking sheets from the oven. Using the back of a 1/2-teaspoon measuring spoon, make an indentation in the center of each warm cookie. Transfer the baking sheets to wire racks and let cool completely.
Fill each cookie with about 1/2 teaspoon of the cold caramel and refrigerate just until set, about 30 minutes.