In a bowl, whisk together flour and salt. Set aside.
Combine sugar, goat cheese, and butter in a large bowl and whisk until mixture is pale and creamy. Whisk in egg yolk and vanilla extract. Fold in the dry ingredients until incorporated and a dough forms. Chill the dough in the refrigerator while you prepare the blackberries.
Mash the blackberries until they are crushed as much as possible. Combine blackberries, sugar, and lemon juice in a small pot and turn the heat to medium. Simmer for 5-8 minutes, stirring continuously, or until berries have cooked into a thick gel. Set aside and let cool. (The jam will thicken as it cools.)
Preheat oven to 350 degrees F. Line a baking sheet with a silicone mat or parchment paper.
Scoop about a tablespoon of dough and roll into a ball. Place the ball on the cookie sheet and press an indentation in the dough to form a hollow for the jelly. Repeat with the rest of the dough, placing the balls about 1-2 inches apart (they will not spread too much).
Place about a half-teaspoon of blackberry jam into each of the dough indentations, or enough to fill it. Combine goat cheese, a splash of milk, and powdered sugar in a pinch bowl. Microwave for 10 seconds to make the mixture slightly thinner and easier to swirl. Place a dot of goat cheese (or as much as desired) into each dough hollow. If you like, you can take a toothpick and run it through the cheese to swirl the jam and cheese together.
Bake for 10 to 12 minutes, or until cookies are set at the edges. The cookies will be pale and soft, but the cookie should be able to lift away from the sheet and the bottom should be just golden. Let cool on the baking sheet for a few minutes, then transfer to a wire rack to cool completely.