Prep the pan and oven: Heat oven to 350°F (175°C). Grease and line a 9x5-inch loaf pan with parchment, leaving a little overhang for easy lifting.
Mix zest with sugar: In a large bowl, combine granulated sugar and lemon zest.
Use your fingers to rub the zest into the sugar until fragrant and slightly damp. This releases the lemon oils.
Whisk in wet ingredients: Add melted butter, oil, yogurt, eggs, lemon juice, vanilla, and milk. Whisk until smooth and glossy.
Combine dry ingredients: In a separate bowl, whisk flour, baking powder, baking soda, salt, and poppy seeds.
Bring it together: Add the dry ingredients to the wet.
Whisk gently or fold with a spatula just until you no longer see dry flour. Do not overmix or the cake can turn dense.
Bake: Pour the batter into the prepared pan and smooth the top. Bake 45–55 minutes, until the top is golden and a toothpick comes out clean or with a few moist crumbs.
Make quick lemon syrup: While the cake bakes, stir together 2 tablespoons lemon juice and 2 tablespoons sugar until the sugar dissolves. Warm briefly if needed.
Brush and cool: Let the cake cool in the pan 10 minutes, then lift it out onto a rack.
Brush the warm cake all over with the lemon syrup. Let it cool completely.
Glaze: In a small bowl, whisk 1 cup powdered sugar with 2–3 tablespoons lemon juice until smooth and pourable. Drizzle over the fully cooled cake.
Let set 15–20 minutes before slicing.