Stir butter, dill, lemon zest, cayenne pepper, salt, and black pepper in a bowl until combined. Set aside.

Fill a large saucepan with 2 to 3 inches of water and bring to a boil over high heat. Reduce heat to medium-low, pour in vinegar and a pinch of salt. Crack an egg into a bowl then gently slip the egg into the water. Repeat with remaining eggs. Poach eggs until whites are firm and yolks have thickened but are not hard, 4 to 6 minutes. Remove eggs from water with a slotted spoon, dab on a kitchen towel to remove excess water, then transfer to a warm plate.