Preheat the oven to 350°F (177°C). Mix the shortening, oil, and flour into a smooth paste and use a pastry brush to work it into every crevice of the crown bundt pan.

Melt the butter in a saucepan over medium heat until it foams, turns clear gold, and smells nutty with brown flecks on the bottom, about 5–7 minutes. Pour into a heatproof bowl and cool 15 minutes.

Whisk together the flour, baking powder, baking soda, and salt in a medium bowl. Set aside.

Whisk the cooled brown butter and sugar together until slightly grainy, about 1 minute. Add the eggs one at a time, whisking well after each, then whisk in the vanilla bean paste.

Add the flour mixture in three additions, alternating with the sour cream and buttermilk, starting and ending with flour. Mix just until no streaks remain.

Pour the batter into the prepared crown pan and tap firmly on the counter twice to release air pockets.

Bake for 50–55 minutes, until a toothpick comes out with a few moist crumbs and the top springs back gently when pressed.

Cool in the pan for exactly 15 minutes, then invert onto a wire rack and lift the pan straight up and off.

Let the cake cool completely, about 1 hour, before glazing.

Whisk the powdered sugar, cream, vanilla bean paste, and salt into a glaze that drips in a slow ribbon. Drizzle over the cooled cake in two thin coats, 10 minutes apart, pressing gold dragées into the wet glaze.

Once the glaze has set, about 20 minutes, place the crown topper into the center hole.