Preheat the oven to 350 degrees F (175 degrees C). Grease a 9-inch round cake pan with cooking spray.
Combine flour, poppy seeds, baking powder, lemon zest, and salt in a small bowl.
Combine butter and sugar in a bowl and beat with an electric mixer until light and fluffy, 2 to 3 minutes. Mix in Greek yogurt. Beat in eggs, one at a time, until thoroughly combined. Add the flour mixture gradually, folding together until just combined. Spoon thick batter into the prepared cake pan and spread evenly.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes. Remove from oven and let cool 10 minutes before transferring to a cake platter or plate to cool completely.
Meanwhile, make the ermine frosting. Combine milk and sugar in a saucepan over medium heat. Whisk until sugar is dissolved, then add flour. Whisk until no lumps remain and the mixture thickens to the consistency of pudding. Remove from heat; stir in lemon juice. Let cool completely, about 10 minutes.
Beat butter and salt with an electric mixer until light and fluffy, about 5 minutes. Gradually add the milk-flour mixture, a tablespoon at a time, and beat on medium speed until well combined and fluffy, 3 to 4 minutes.
Spread the frosting over the cooled poppy seed cake.