Preheat the oven to 350°F. Grease an 18 by 13 inch half sheet pan or line it with parchment paper.
Melt butter and water in a saucepan over medium-high heat. Stir in the brown sugar and bring the mixture to a full rolling boil.
Remove the saucepan from the heat and cool for a couple of minutes. Stir in the vanilla extract.
Add the flour, salt, and baking soda. Whisk until the batter is smooth.
Whisk the eggs and sour cream in a separate bowl, then stir them into the batter until just combined.
Spread the batter in the prepared pan. Bake for 17 to 20 minutes, until the center is springy and a toothpick comes out without wet batter.
For the icing, melt the butter in a clean saucepan. Add the brown sugar, evaporated milk, and salt.
Bring the icing to a rolling boil over medium heat, stirring constantly, until it coats the back of a spoon.
Remove from the heat and stir in the vanilla. Place the saucepan in an ice bath and stir for 3 to 5 minutes, until thick but still pourable.
Pour the icing over the slightly cooled cake and spread it to the edges. Let it set briefly before slicing.