Preheat oven to 350 degrees F. Line two muffin pans with cupcake liners or silicone baking cups.

In a medium bowl sift together flour, salt, baking powder and baking soda.

In the bowl of an electric mixer, cream butter and sugar until pale and fluffy, about 5 minutes on medium speed. Add the lime zest and mix for an additional minute. Add eggs one at a time, beating until combined after each addition. Beat in vanilla extract and lime juice.

Reduce mixer speed to low and slowly add about half of the flour mixture. Add half of the milk, then the rest of the flour and the rest of the milk. Beat until just combined. Scrap down the side of the bowl as needed.

Pour batter into prepared pans. Fill liners about ⅔ full. Bake cupcakes until lightly golden or a toothpick inserted into the center comes out clean, about 12-14 minutes.

Remove cupcakes from pan and let cool completely on wire rack before frosting.

Make compote first: Combine all the ingredients for compote plus ½ cup water in a small saucepan set over high heat. Bring to a boil and reduce heat to medium. Cook, stirring occasionally, until berries are soft and thickened, about 15 minutes. Remove from heat. Pass through a fine sieve if desired. Let cool to room temperature before using.

Whip butter for 8 minutes on medium speed using the paddle attachment of an electric mixer. The butter will become pale and creamy.

Add the rest of the ingredients, except blackberry compote and mix on low speed for 1 minute. Then increase speed to medium and mix for 6 minutes until the frosting is very light, creamy and fluffy. Add blackberry compote and whip for additional minute or two to combine.

Frost cooled cupcakes.