Preheat oven to 350 degrees F. Grease and flour a 9x5-inch loaf pan; set aside.
In a medium bowl, whisk together the flour, baking powder, and salt; set aside.
Add the sugar and lemon zest to the bowl of a food processor; process until combined, about five 1-second pulses. Add the eggs, lemon juice and vanilla; process until combined, about 5 seconds. With the machine running, add the melted butter through feed tube in a steady stream. Transfer the mixture to large bowl. Sift the flour mixture over eggs in three additions, whisking gently after each addition until just combined.
Pour the batter into the prepared pan and bake for 15 minutes. Reduce the oven temperature to 325 degrees F and continue to bake until deep golden brown and a skewer inserted into the center comes out clean, about 35 minutes, rotating the pan halfway through baking time. Cool in pan on a wire rack for 10 minutes.
Meanwhile, bring the sugar and lemon juice for the glaze to a boil in a small saucepan, stirring occasionally to dissolve the sugar. Reduce the heat to low and simmer until thickened slightly, about 2 minutes.
Remove the cake from the pan and place on a wire cooling rack. Using a toothpick, poke holes all over the top and sides of the cake. Brush the lemon glaze all over the top and sides of the cake. Allow the cake to cool to room temperature, at least 1 hour. The cake can be stored at room temperature, wrapped tightly in plastic wrap, for up to 5 days.