Preheat the oven to 375°F. Spray your muffin tin with cooking spray.
Melt butter and let it cool.
In a large bowl, mix the sugar and the lemon zest together with your fingertips until the sugar is moist. Whisk in the flour, baking powder, baking soda and salt. In a small bowl, whisk the Greek yogurt, eggs, vanilla, lemon juice and melted butter together until well blended.
Pour the liquid ingredients over the dry ingredients and gently stir to blend. Don't overmix. Stir in the poppy seeds. It's going to seem really thick..don't worry, it's going great! Divide the batter evenly among the muffin tin.
Bake for 14-18 minutes, or until the tops are golden and a thin knife inserted into the center of the muffins comes out clean. Transfer the pan to a rack and cool for 5 minutes before carefully removing each muffin from its mold. Cool the muffins completely on the rack before icing them.
While muffins are baking, mix up the glaze. In a small bowl, stir the confectioner's sugar and lemon juice. If glaze needs to be further thinned, as a tiny bit of lemon juice at a time until it's a drizzle-able consistency.