Heat the oven to 350°F. Coat an 8-by-4-inch loaf pan with cooking spray.
In a large bowl, whisk together the yogurt, eggs, lemon zest, vanilla extract, poppyseeds, and 1 cup of the granulated sugar. In a small bowl, whisk together the flour, baking powder, and salt. Slowly stir the dry mixture into the wet mixture. Gently fold in the oil.
Pour the batter into the prepared pan and bake until a toothpick inserted in the center of the cake comes out clean, 45 to 55 minutes. Cool the cake in the pan for 10 minutes and then remove it to a rack. Place a sheet of foil underneath the rack.
Combine 4 tablespoons of the lemon juice and the remaining ¼ cup of granulated sugar in a small saucepan. Cook over medium heat until the sugar dissolves. Brush and dab the mixture over the bottom, sides, and top of the warm cake, letting it soak in. Make sure the cake is right side up to cool completely.
When it is cool, stir together the confectioners’ sugar and remaining 1 tablespoon of lemon juice. Drizzle the topping over the cake.
Tip: You’ll need two lemons for this cake. Zest the first lemon; then squeeze both lemons for the juice.