Make the jam first so it can cool: combine halved cherry tomatoes, sugar, vinegar, diced onion, minced garlic, crushed red pepper, ground cumin, and 1/2 tsp kosher salt in a medium saucepan.
Bring the tomato mixture to a steady simmer over medium heat, stirring occasionally. As the tomatoes soften, mash them gently against the side of the pan with a wooden spoon to break them down.
Maintain a simmer and cook until the mixture reduces and thickens to a jammy consistency, about 15–20 minutes (times may vary). Remove the pan from the heat, stir in the bourbon, then let the jam cool to room temperature before chilling; refrigerate at least 2 hours or overnight.
While the jam cools, prepare the cookie dough: in a large bowl, beat the softened butter with the granulated sugar and 1/2 tsp kosher salt until smooth and slightly airy, about 1–2 minutes.
Stir in the egg and the lemon zest until just combined. Fold in both shredded cheeses, chopped jalapeño, and chopped rosemary so they’re evenly distributed in the butter mixture.
In a separate bowl whisk together the flour and semolina. Add the dry mix to the cheese-butter mixture in two additions, folding with a sturdy spatula until a soft, slightly sticky dough forms. If needed, finish bringing it together with lightly floured hands.
Cover the dough and chill in the refrigerator for at least 30 minutes (or up to overnight). Chilling firms the butter and makes shaping easier.
Preheat the oven to 350°F. Line baking sheets with parchment paper. Using a small cookie scoop or spoon, portion dough into 1 1/4-inch balls (about 28–32 g each) and arrange 2 inches apart on the prepared sheets.
Make a well in each dough ball using the back of a small spoon or a slightly floured thumb to create a generous indentation that will hold the jam. Return the trays to the fridge for 10–15 minutes while the oven finishes preheating to keep the cookies from spreading too much.
Remove trays from the refrigerator and spoon about 1 teaspoon of chilled bourbon tomato jam into each indentation. Bake one sheet at a time (or two, rotating halfway) for 18–22 minutes, until the cookie edges are lightly golden and the bottoms are set.
Let the cookies cool on the baking sheet for 5 minutes so the jam sets, then transfer to a wire rack to finish cooling. Do not stack cookies while warm—the jam will stick.
Serve slightly warm or at room temperature. Cookies keep in a single layer in the fridge for up to 5 days; bring to room temperature before serving for best texture.