Prepare the Doughnuts: Beat brown sugar, granulated sugar, and butter in bowl of a heavy-duty stand mixer fitted with paddle attachment on medium speed until crumbly, about 2 minutes. Add eggs, 1 at a time, beating just until combined after each addition. Beat in vanilla.

Stir together flour, baking powder, baking soda, and salt in a large bowl. Stir together Slow-Cooker Apple Butter and buttermilk in a 2-cup glass measuring cup. Beating on low speed, gradually add flour mixture to brown sugar mixture alternately with apple butter mixture, beginning and ending with flour mixture until fully incorporated. Cover and chill at least 2 hours or up to 24 hours.

Turn dough out onto a lightly floured work surface, and knead 5 or 6 times. Roll dough to ¼-inch thickness, and cut with a 2 ½-inch round cutter. Cut holes out of center using a 1-inch round cutter. Reroll all scraps once; repeat cutting process.

Heat 2 inches of oil in a large Dutch oven to 375°F; fry 4 or 5 Doughnuts at a time (do not overcrowd) until browned and done, 1 to 2 minutes per side. Transfer to a baking sheet lined with paper towels.

Prepare the Salted-Caramel Glaze: Bring brown sugar, butter, and milk to a boil in a small saucepan over medium, stirring constantly. Boil, stirring constantly, 1 minute. Remove from heat, and stir in vanilla and kosher salt. Stir constantly 2 minutes; use immediately.

Dip top half of Doughnuts in Salted-Caramel Glaze, and sprinkle with flaky sea salt. Let stand 10 minutes; serve warm.