Preheat oven to 325°F with rack in middle. Butter and lightly flour an 8 1/2- by 4 1/2-inch loaf pan.
Heat butter in a 10-inch heavy skillet over medium heat until milk solids on bottom are a dark chocolate brown. Transfer to a shallow bowl and chill in freezer until just congealed, about 15 minutes.
Whisk together flour, baking powder, and salt.
Beat together brown butter and sugars with an electric mixer until fluffy, about 2 minutes. Add eggs 1 at a time, beating well after each addition. Beat in vanilla and then the bourbon (!). At low speed, mix in flour mixture until just incorporated.
Transfer batter to pan, smoothing top, then rap pan on counter to settle batter. Bake until golden-brown and a wooden pick inserted into center comes out clean, 1 to 1 1/4 hours.
Cool in pan 30 minutes, then invert cake onto a rack and cool completely, right side up, 1 hour.