Preheat the oven to 350F.
In a large bowl, beat the butter and sugar until pale and creamy, about 3 to 4 minutes. Beat in the egg and vanilla.
Put the rhubarb into a bowl - Elizabeth says if it's frozen, to thaw and drain it, but I used frozen and kept it frozen - and sprinkle over a spoonful of the flour, shaking to coat the rhubarb. Put the rest of the flour into a small bowl or measuring cup and stir in the baking soda and salt. Add to the butter mixture in three additions, alternating with the buttermilk in two.
Spread the batter into a parchment-lined 9x13-inch baking pan and sprinkle with the rhubarb. Press the rhubarb down with the back of your spatula to help it sink in a bit. (I did this because I missed the stirring it into the batter part - you could do that instead, if you like.)
In a small bowl, mix together the brown sugar, cinnamon and butter with a fork and/or your fingers, and sprinkle the mixture over the top of the cake. Bake for 45 minutes, or until deep golden and the cakey parts are springy to the touch. Elizabeth says a toothpick inserted into the centre will come out clean, but I think that's debatable depending on where the sunken bits are - if you test this way, go for a spot near the middle that's clear of rhubarb, because the finished cake is particularly dense and sticky.
Let cool or serve warm, with vanilla ice cream. Serves 10.