Combine bourbon and mint in a small bowl and let sit for 15 minutes.After 15 minutes, pour trough strainer. Reserve the mint for the cake batter.
Combine flour, salt, baking powder, and baking soda in a medium bowl. Set aside.
In a glass measuring cup, combine buttermilk, vanilla, and 1/4 cup of the mint-infused bourbon, plus the reserved mint.
Using an electric stand mixer, beat butter and sugar until light and fluffy, about 5 minutes.
Add the eggs and egg yolk one at a time, beating each one until mixed in and stopping a few times to scrape down the sides of the bowl.
With mixer on low, add 1/3 of flour mixture. Once mixed in, add 1/2 of buttermilk mixture. Repeat with another 1/3 of flour, remaining buttermilk, and remaining flour. Stop to scrape down the sides a few times while mixing in the flour and buttermilk.
Pour batter into a 12-cup Bundt pan or tube pan that has been sprayed well with baking spray.
Bake in a 350 degree oven for 50 to 60 minutes or until it feels firm when you press down on the top. Let cool 10 minutes in pan and then invert onto a cake stand.
To make Brown Sugar Syrup, heat brown sugar and water in a small saucepan. Stir to dissolve the sugar. Once sugar is dissolved, remove from heat and stir in 2 tablespoons of the mint-infused bourbon. Brush syrup over top and sides of cake.Let cake completely cool before glazing.
To make glaze, whisk together powdered sugar, melted butter, food coloring and last 2 tablespoons of bourbon. Drizzle over cake.