Adjust an oven rack to the center position and preheat the oven to 325°F (165°C). Generously butter a 10-inch bundt pan, then dust it with flour, tapping out any excess. Set the pan aside.
In the bowl of a stand mixer fitted with the paddle attachment (or using a large bowl and a hand mixer), beat the 1 cup of cubed butter with 1 3/4 cups sugar on medium-high until the mixture is pale and fluffy, about 3 to 4 minutes.
Add the eggs one at a time, beating briefly after each addition so they incorporate smoothly. Mix in 1 tablespoon vanilla and the lemon zest until evenly distributed.
In a separate bowl whisk together the flour, baking powder, baking soda, and kosher salt. Scoop about one-third of the dry mix into the creamed butter mixture and blend on low just until combined.
Pour in half of the buttermilk and mix on low until mixed, then add another third of the dry ingredients. Repeat with the remaining buttermilk and end with the final third of flour, mixing only until the batter is smooth and no streaks of flour remain. Avoid overmixing.
Scrape the batter into the prepared bundt pan and smooth the top with a spatula. Tap the pan gently on the counter to remove any large air bubbles.
Bake the cake in the preheated oven for 60 to 70 minutes, or until the top is golden and a thin skewer or toothpick inserted near the center comes out with just a few moist crumbs attached. Rotate the pan halfway through baking if your oven has hot spots.
While the cake bakes, make the glaze: combine 1/3 cup butter, 3/4 cup sugar and 2 tablespoons water in a small saucepan over low heat. Stir constantly just until the butter melts and the sugar dissolves (do not boil). Remove from heat and stir in 2 teaspoons vanilla.
As soon as the cake comes out of the oven, let it rest 10 minutes in the pan. Then use a skewer or thin knife to poke holes all over the warm cake so it will absorb the glaze.
Slowly pour the warm glaze over the cake while it is still in the pan, allowing the liquid to sink in before adding more. Reserve any glaze left in the pan and spoon it over the cake again after it sits for 10 minutes.
Let the cake cool in the pan for about 30 to 40 minutes to set and absorb the glaze. Carefully invert onto a serving plate and cool completely on a rack. For best texture and melded flavor, cover and rest the cake overnight before slicing.