For the cake: Preheat the oven to 350 degrees F. Adjust the oven rack to the center of the oven. Grease two loaf pans with butter and dust with flour.
Add the butter and sugar to the bowl of a stand mixer. Using the paddle attachment, beat until light and fluffy, stopping once or twice to scrap down the sides of the bowl with a rubber spatula, about 5 minutes. Add the eggs, one at a time, beating well after each addition; then add the clementine juice and zest and the vanilla.
In a medium bowl, whisk together the flour, baking powder, salt and cinnamon to get rid of any lumps. Add the flour to the mixer in two additions, alternating with the milk, and mixing until combined.
Pour the batter into the prepared loaf pans. Bake for 1 hour 10 minutes, until golden and light. Let the cakes cool for 15 minutes before turning them out of the pans. Set the cakes right-side-up on a rack to cool.
For the glaze: In a bowl, combine the confectioners' sugar and the clementine juice and zest. Whisk together until smooth. Apply the glaze to the cakes while they are still warm.