In a medium bowl, toss the chopped strawberries with 2 teaspoons of the sugar and 1/4 cup water. Let stand at room temperature for 1 hour until the berries release their juices and measure about 1 1/2 cups. Pulse the berry mixture in a blender or food processor on low speed just until chunky, not smooth. Reserve 1/2 cup of the crushed strawberries for the frosting; set the rest aside for the cake batter.

Preheat the oven to 400 degrees F.

In a large bowl, sift together the remaining 2 cups sugar and the all-purpose flour.

Melt the 2 sticks of butter in a small saucepan over low heat. In another bowl, whisk the melted butter with the eggs, buttermilk, vanilla extract, and baking soda. Pour the liquid into the flour mixture and stir until just combined. Fold in 1 cup of the reserved crushed strawberries.

Pour the batter into a greased 9x13-inch baking pan. Bake for 20 to 25 minutes, until a knife inserted into the center comes out clean.

While the cake bakes, make the frosting: In a mixing bowl, beat the softened butter and cream cheese together until smooth. Gradually add the powdered sugar, beating until fully incorporated. Mix in the reserved 1/2 cup crushed strawberries, lime juice, lime zest, and vanilla extract.

Let the cake cool completely on a wire rack, then spread the strawberry cream cheese frosting evenly over the top.