Heat the oven to 350℉. Grease and flour a (8-x-5-x-2½-inch) loaf pan. Line the bottom with parchment, trimmed to extend over 2 opposing sides by 1 inch. This will make a sling to help lift the cake out of the pan. Grease and flour the parchment.
Make the lemon syrup: Whisk the sugar and juice together in a small bowl. Return periodically to re-whisk so all sugar dissolves by the time the cake has baked.
Make the cake: In the bowl of an electric mixer, rub the zest into the sugar with your fingertips until the sugar is slightly damp and fragrant with lemon.
Cream the lemony sugar with the butter and vanilla on high speed with a paddle attachment until fluffy, about 5 minutes. Reduce the speed to medium and add the eggs, one at a time, beating thoroughly after each addition.
Sift the flour, baking powder, baking soda and salt together into a large bowl. Always mixing on low speed, add the flour mixture and yogurt alternately to the butter-sugar mixture in 5 installments, starting and finishing with the flour. When the last trace of flour disappears, stop the mixer. Scrape the interior of the bowl with a rubber spatula, folding the batter a few times to ensure it is evenly mixed.
Scrape the batter into the pan, smooth the top, and bake for 45 to 55 minutes, or until the cake's top is golden, a cake tester comes out clean and the cake's internal temperature is 210℉. Remove from the oven to a rack, leaving the cake in the pan.
Soak the cake: After 5 minutes, use a thin skewer or toothpick to poke many small holes in the cake's surface, all the way down into the cake. Pour the lemon syrup all over the top of the hot cake. Let cool in the pan for an hour before carefully lifting it out. Wrap it snugly in plastic wrap. Let sit until fully cooled, overnight if you can.
Glaze the cake: Unwrap the completely cooled cake and set it on a wire rack over a sheet pan or piece of parchment.
Whisk the confectioner's sugar and 1½ Tbsp lemon juice in a bowl until smooth. Adjust with more juice as needed, so the glaze slowly drizzles down the side of the cake when you pour it on. Let set for at least an hour before serving.