Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13 inch baking pan.
In a large bowl, cream together 1/2 cup butter and the turbinado sugar. Add the eggs, and beat for 1 minute. Stir in the vanilla and salt.
In a small bowl, stir together the sour cream and the baking soda until the baking soda has dissolved. Add the sour cream mixture and the flour to the egg mixture. Beat well, then stir in the mashed strawberries.
Pour the batter into the prepared pan. Bake in preheated oven for 30 minutes, or until top springs back when lightly touched.
While cake is cooling, make the icing: Cream together 2 tablespoons butter and confectioners' sugar. Add the milk, lemon juice, and lemon zest, and beat until smooth. Spread icing on warm (not hot!) cake, and sprinkle with slivered almonds.