Make the pound cake: Preheat the oven to 350˚. Butter the bottom and sides of a 9-by-5-inch loaf pan, then line with parchment paper in two overlapping strips, leaving an overhang. Butter the parchment and dust with flour, tapping out the excess.
Grate the zest of 1 1/2 lemons and remove the remaining zest in wide strips with a vegetable peeler. Juice both lemons into a small bowl; set aside.
Whisk the eggs, egg yolk, 1/4 cup heavy cream and 2 teaspoons vanilla in a medium bowl until slightly foamy; set aside. Beat the butter and 1 cup cane sugar in a large bowl with a mixer on medium speed until well combined, 1 to 2 minutes. Scrape down the bowl, then beat on medium-high speed until pale and fluffy, 6 to 7 minutes, adding the grated lemon zest after about 5 minutes. Reduce the mixer speed to low and beat in the flour until just combined. Gradually add the egg mixture and beat until combined. Scrape down the bowl; increase the speed to medium and beat until fluffy, 15 to 30 seconds.
Transfer the batter to the prepared pan and smooth the top. Bake until golden and the top springs back when gently pressed, 1 hour 10 minutes to 1 hour 20 minutes.
Meanwhile, make the lemon syrup: Bring a small saucepan of water to a boil. Add the lemon zest strips and cook 1 minute; drain. Repeat twice, using fresh water each time; drain and transfer the lemon zest to a cutting board. Wipe out the saucepan.
Thinly slice the lemon zest strips and return to the saucepan along with the reserved lemon juice, the remaining 1/4 cup cane sugar and 1/4 cup water. Bring to a simmer over medium heat, stirring to dissolve the sugar; cook until slightly reduced, 3 to 5 minutes. Let cool. Strain the syrup through a fine-mesh sieve; reserve the syrup and candied zest.
Remove the cake from the oven, transfer to a rack and let cool 10 minutes in the pan. Poke 12 holes in the top of the cake, then brush with half the lemon syrup. Let soak 5 minutes, then lift the cake from the pan and remove the parchment. Transfer to a rack. Poke a few holes in the sides of the cake, then brush all over with the remaining lemon syrup. Let cool completely.
Make the glaze: Combine the confectioners' sugar, 3 tablespoons heavy cream and the remaining 1/4 teaspoon vanilla in a medium bowl and whisk until thick and smooth; thin with the remaining 1 tablespoon heavy cream, if needed. Spoon the glaze over the cooled cake and top with the candied lemon zest. Let set at least 20 minutes.
Quality ingredients make a big difference in pound cake. Try using premium butter — and if you can, use organic cane sugar. It gives the loaves a great crunchy top.