In a bowl, mix flour, poppy seeds, lemon peel, baking powder, and salt. In a small bowl, whisk eggs, sugar, buttermilk, melted butter, and vanilla to blend. Stir into dry ingredients until well blended. Chill until firm, at least 1 hour or up to 3 hours.
Scrape dough onto a generously floured surface. With floured hands, roll dough into 1-inch balls. Place doughnut holes on a floured baking sheet.
Meanwhile, fill an electric deep-fryer to the fill line or pour about 4 inches of oil into a 5- to 6-quart pan; heat to 375°. Using a slotted spoon to pick up one or two doughnut holes at a time, gently lower three or four holes into oil and cook, turning once, until golden brown, 3 to 4 minutes (to check timing, cut first doughnut hole to test). Using slotted spoon, transfer doughnut holes to paper towels to drain. Repeat to fry remaining holes.
When cool enough to handle, roll each hole in glaze to coat. Place on a plate and let stand until glaze is set, about 2 minutes.
Lemon glaze. In a bowl, mix 1/4 cup lemon juice and 2 cups powdered sugar until smooth.