Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan, line the bottom with a parchment circle, then flour the pan and tap out the excess.
Whisk the flour, baking powder, and salt in a medium bowl.
In a large bowl, rub the lemon zest into the sugar with your fingertips until fragrant and pale yellow. Whisk in the yogurt, oil, eggs, and vanilla until smooth.
Fold the dry ingredients into the wet ingredients until just combined (do not overmix).
Scrape the batter into the prepared pan and smooth the top. Bake 45–50 minutes until deeply golden and a toothpick inserted in the center comes out clean.
For the syrup: In the last 5–10 minutes of baking, heat the sugar and lemon juice in a small saucepan over medium heat, stirring just until dissolved. Do not boil.
While the cake is hot in the pan, poke holes all over the surface with a skewer or toothpick. Slowly pour the warm syrup evenly over the cake. Cool 20 minutes in the pan, then invert onto a wire rack and cool completely.
For the glaze: Whisk the powdered sugar and lemon juice until thick but pourable. Transfer the cooled cake to a platter, optionally arrange the thin lemon slices on top, and drizzle with glaze. Let set 15–20 minutes before slicing.
