Preheat the oven to 350°F (175°C). Grease an 8.5x4.5-inch loaf pan and line it with parchment paper, leaving an overhang on the sides for easy removal.
Add granulated sugar and fresh lemon zest to a large bowl, and rub them together using your fingertips for 1 minute until the sugar is fragrant and moist.
Whisk plain whole milk yogurt, vegetable oil, fresh lemon juice, and pure vanilla extract into the scented sugar until completely blended and smooth.
Sift all-purpose flour, baking powder, baking soda, and salt into the wet mixture, add poppy seeds, and stir gently with a spatula just until incorporated without overmixing.
Spread the thick batter evenly into the prepared loaf pan and smooth the top surface with a spatula.
Bake at 350°F (175°C) for 45 to 50 minutes until a wooden skewer inserted into the center comes out clean and the loaf is golden on top.
Cool the cake in the pan for 15 minutes, then use the parchment paper handles to lift the loaf onto a wire rack to cool completely.
Whisk powdered sugar, fresh lemon juice, and lemon zest together in a small bowl until smooth, then drizzle the thick glaze over the cooled loaf and allow it to set before slicing.
