Preheat the oven to 350 F degrees. Measure all ingredients. Take butter out to soften. Zest and juice limes. Grease a 4-quart cake tube or bundt pan (see notes for other pan types and cooking times).
Whisk together the flour, salt, baking powder, and baking soda. Set aside.
Using a hand mixer or stand mixer on medium speed, beat the butter and sugar until light and fluffy. Add the eggs, one at a time, mixing well after each. Mix in the vanilla, lime juice, and lime zest.
Reduce mixer speed to low. Alternate adding and mixing in the flour and the milk... ⅓ of the flour, ½ milk, ⅓ flour, ½ milk, last ⅓ of the flour.
Transfer the batter to the prepared pan, and spread evenly. Bake 50 to 60 minutes, or until a pick inserted into the center comes out clean.
Cool the cake in the pan for 15 minutes. Then, gently run a knife around the outside of the cake inside the pan and flip the cake out into your hands and gently place it right side up onto a serving platter e to continue cooling.
Make the Lime Glaze: In a saucepan, add the butter, sweetened condensed milk, and the lime juice and zest. Over low heat, stir to combine as the butter melts and everything comes together. Do not overheat, as the glaze can start to clump on the bottom of the pan. Remove from the heat and let cool. It can still be warm when poured over the cake, just not hot.
Pour over the cake and let it drip down the sides. You can pour as much or as little as you like and save the rest to warm up later and let people pour over their slice.