Preheat oven to 350 degrees. Spread oats on a rimmed baking sheet; bake until golden, about 10 minutes. Transfer to a medium bowl; stir in flours and salt. In a large bowl, using an electric mixer, beat butter and brown sugar on medium until fluffy. Add vanilla; beat to combine. Scrape down bowl. Add oat mixture; beat until blended. Form dough into a disk, wrap in plastic, and refrigerate until firm, 2 hours.
Preheat oven to 325 degrees, with racks in upper and lower thirds. On a lightly floured surface, roll dough 1/8 inch thick, dusting lightly with flour as needed. Cut out cookies with a 2-inch heart-shaped cookie cutter; place 1/2 inch apart on two parchment-lined baking sheets. Bake until cookies are golden around edges, 12 to 15 minutes, rotating sheets halfway through. Let cool 5 minutes, then transfer cookies to a wire rack to cool completely. Repeat with remaining dough, chilling if too soft (reroll and cut scraps). Spread filling on flat side of half the cookies; top with remaining cookies.