Preheat oven to 350°F (180°C). Line 2 baking sheets with parchment paper.

In a large mixing bowl, combine sugar, salt, and lemon zest. Rub together with your fingertips until fragrant and the sugar looks slightly moist.

Add the softened butter and beat with an electric mixer until light and fluffy, about 2 minutes.

Add the oil, vanilla, and lemon juice. Beat until smooth.

Sift the flour and baking soda into the bowl. Stir until just combined. Fold in poppy seeds.

Scoop about 2 tablespoons of dough per cookie onto prepared baking sheets, spacing 2 inches apart. Flatten each slightly with your fingertips.

Bake for 9–11 minutes, until the edges are set but the centers still look soft. Let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack.

In a small bowl, whisk powdered sugar and lemon juice until smooth. Spoon or drizzle glaze over cooled cookies. Let set for 10 minutes before serving.