Use a high-quality bourbon -- we like Bulleit -- for the glaze; the flavor will shine through.
Preheat oven to 350 degrees. Brush a standard Bundt pan with butter, then dust with flour, tapping out excess.
Melt butter in a saucepan over medium-high. When it boils, reduce heat to medium; simmer until foamy. Continue cooking, stirring occasionally and scraping bottom of pan, until foam subsides, butter turns golden brown with a nutty aroma, and milk solids separate into brown specks that sink to bottom, 2 to 7 minutes. Remove from heat; transfer to a heatproof bowl and let cool 10 minutes. Whisk together flour, baking powder, baking soda, and salt in a bowl. In another bowl, whisk together eggs, sugar, buttermilk, and vanilla. Whisk brown butter into egg mixture, then stir in flour mixture just to combine (do not overmix). Pour batter into prepared pan; smooth top with a spatula. Bake until golden brown and a toothpick inserted into center comes out clean, 50 to 55 minutes. Let cool on a wire rack 15 minutes.
Melt butter in a saucepan over medium-high. When it boils, reduce heat to medium; simmer until foamy. Continue cooking, stirring occasionally and scraping bottom of pan, until foam subsides, butter turns golden brown with a nutty aroma, and milk solids separate into brown specks that sink to bottom, 2 to 7 minutes. Remove from heat; carefully add bourbon (mixture will bubble vigorously). When bubbles subside, stir in sugar to combine. Cook over medium-low, stirring occasionally, just until mixture is smooth and runs from a spoon in a thin, translucent stream (sugar will not dissolve completely; do not let mixture boil). Pierce holes in cake (still in pan) to bottom with a wooden skewer at 1-inch intervals. Slowly drizzle glaze evenly over cake; let stand until glaze is fully absorbed, about 15 minutes. Invert onto a cake plate. Let cool completely before slicing and serving. Cake can be stored in an airtight container up to 3 days.