Heat the oven to 325°F (165°C). Thoroughly grease and flour a 10-inch Bundt or tube pan, or coat it with baking-release spray containing flour.

Whisk the flour, baking powder, and fine salt together in a medium bowl.

Beat the softened butter and granulated sugar on medium speed for 4 to 5 minutes, until very light and fluffy. Scrape the bowl once or twice so the mixture creams evenly.

Add the room-temperature eggs one at a time, beating well after each addition. Scrape down the bowl as needed before adding the next egg.

Mix in the vanilla bean seeds, vanilla bean paste, and pure vanilla extract until evenly distributed.

Add the dry ingredients in three additions, alternating with the milk in two additions. Mix only until the batter is smooth, taking care not to overmix after the flour goes in.

Transfer the batter to the prepared pan and smooth the surface. Tap the pan lightly on the counter to remove large air pockets.

Bake for approximately 75 to 90 minutes. Begin checking around 70 minutes, and remove the cake when a skewer inserted into the thickest part comes out clean or with a few moist crumbs.

Cool the cake in the pan for 10 to 15 minutes, then carefully invert it onto a cooling rack. Let it cool completely, allowing about 2 hours total cooling time before glazing.

Whisk the confectioners' sugar, 2 to 3 tablespoons of milk or heavy cream, vanilla extract, and a pinch of salt until pourable. Drizzle the glaze over the cooled cake and allow it to set before slicing.