Preheat your oven to 325°F. Generously grease and flour a 10- or 12-cup bundt pan, or use a baking spray with flour (such as Baker's Joy)..
In the bowl of a stand mixer fitted with the paddle attachment (or a large mixing bowl with an electric hand mixer), beat the softened butter and sugar on medium-high speed for 3–5 minutes until light and fluffy.
Add eggs one at a time, beating well after each. Scrape down the sides of the bowl as needed.
Mix in the dry vanilla pudding mix and vanilla extract until fully incorporated.
In a separate bowl, whisk together the flour, baking powder, and salt.
Alternate adding the flour mixture and milk to the wet ingredients, beginning and ending with flour. Mix until just combined—don’t overmix.
Pour the batter into your prepared Bundt pan and smooth the top.
Bake for 60–70 minutes, or until a toothpick inserted in the center comes out clean.
Cool in the pan for 15 minutes, then carefully run a knife along the edge of the cake to ensure it's loosened from the pan. Then turn out onto a wire rack to cool completely.
Whisk together the powdered sugar, milk, and vanilla extract until smooth. Drizzle over the cooled cake.