In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugar at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition and stopping to scrape sides of bowl. (Mixture may look a bit broken, but batter will come together.)
In a medium bowl, whisk together flour, baking powder, and salt. In a small bowl, whisk together buttermilk and vanilla. With mixer on low speed, gradually add flour mixture to butter mixture alternately with buttermilk mixture, beginning and ending with flour mixture, beating just until combined after each addition and stopping to scrape paddle and bottom and sides of bowl. Cover and refrigerate for at least 1 hour.
Line 2 baking sheets with parchment paper.
Transfer dough to a heavily floured surface. Fold dough on itself once just to bring it together; pat or roll to ⅜- to ½-inch thickness. Using a 3½-inch doughnut cutter dipped in flour, cut dough, rerolling scraps and reflouring surface as necessary. Gently transfer cut dough to prepared pans.
In a 4- to 6-quart cast-iron Dutch oven, pour oil to a depth of 2 inches, and heat over medium heat until a deep-fry thermometer registers 350°F (180°C). Line a large rimmed baking sheet with paper towels.
Fry doughnuts, 2 at a time, until golden brown, 2 to 3 minutes per side, adjusting heat as needed to maintain temperature. Remove from oil using a spider strainer or slotted spoon, and let drain on prepared pan. Let cool slightly.
Place doughnuts on a wire rack; spoon Bourbon-Braised Pineapple Glaze onto doughnuts. Let stand until glaze is set, 5 to 10 minutes. Best served same day as made.