In a medium bowl, whisk together the rolled oats, all-purpose flour, baking powder, and salt to evenly distribute the dry ingredients.
In a large mixing bowl, cream the softened unsalted butter with the brown sugar and granulated sugar using an electric mixer or whisk until light and fluffy, about 3 to 4 minutes.
Stir in the vanilla extract into the creamed butter and sugar mixture.
Gradually add the dry oat and flour mixture into the wet ingredients, mixing on low speed or folding gently until the dough comes together, thick and slightly crumbly but holding shape.
Portion dough balls roughly 1 tablespoon in size onto a baking sheet lined with parchment paper, spacing about 2 inches apart. Press an indentation into the center of each dough ball using your thumb or the back of a teaspoon.
Spoon about 1/2 teaspoon of blackberry jam into each indentation, being careful not to overfill.
Bake in a preheated oven at 350°F (175°C) for 12-15 minutes, until edges are golden and jam is set but not bubbly.
Remove cookies from oven and cool on baking sheet for 5 minutes. Transfer to a wire rack to cool completely before serving.