Preheat the oven to 350°F (180°C). Grease a 9x13 (or slightly smaller) baking dish, and line it with baking parchment.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside.
Beat the butter and sugar until light and fluffy -- about 4 minutes. Beat in the egg. A little at a time, beat in the milk, followed by the lemon zest. With the mixer at its lowest speed, gradually beat in the flour. Increase the speed to "medium-high," and beat just until the batter appears smooth -- 5-10 seconds.
Pour half of the batter into the prepared baking dish, and spread it all the way to the edges. (The layer will be quite thin.) Sprinkle with half the blueberries. Top with the remaining batter, and sprinkle on the remaining blueberries.
Bake until the cake turns golden, and a skewer inserted into the center comes out clean -- 35-45 minutes (my electric oven required the full 45 minutes).
Allow the cake to cool in the pan for 5 minutes. Then transfer the cake on its parchment liner to a wire rack, and let it cool to room temperature.
To make the icing, whisk together the confectioners' sugar and enough lemon juice to achieve a thick but pourable glaze. Drizzle the icing over the cake.
For easy slicing, remove the cake from its parchment liner (run an offset spatula between cake and paper if necessary to dislodge), and place it on a cutting board. Cut the cake into 12 rectangles.