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Mince the basil finely and then add it to the milk. Set it aside and let it steep for 10 minutes at a minimum, to let the basil oil mix with the milk.
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Preheat the oven to 350 F. Butter and flour a 9 inch, 10 cup bundt pan. Set aside.
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Whisk together the flour, salt and baking powder. Set aside.
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Beat the butter and lime zest in an electric mixer until creamy, about 2 minutes. Slowly add the sugar with the mixer on medium speed, and beat until fluffy, about 2 minutes more. Add the eggs one at a time, beating well after each one. Beat in the vanilla and lime juice.
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Alternating ingredients, with the mixer on low speed, mix in the flour mixture and basil milk in 4 and 3 additions, respectively, beginning and ending with the flour mixture. Pour the cake batter into the prepared pan and smooth the top.
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Bake for about 40 minutes, you will know the cake is ready to come out when a cake tester comes out clean or with only a few crumbs attacher, and the sides of the cake have begun to pull away from the pan. Let the cake cool in the pan for 15 minutes.
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While the cake is cooling, make the glaze. I like a tart glaze, so I used the remaining lime juice from the 4 limes. Mix this with a cup of sugar and bring to a boil in a small saucepan. Taste, and if you want it sweeter, add more sugar. Let boil for 5 minutes and then remove from the he
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After inverting the cake onto a cooling rack, brush the warm cake with the warm glaze. I brushed on about 3 layers; the tart-sweet lime glaze is my favorite part so I like a good layer of it! Cool before slicing.