Bring a saucepan of lightly salted water to a boil. Add chicken breasts; cook until no longer pink and an instant-read thermometer inserted into the center reads at least 165 degrees F (74 degrees C), about 20 minutes. Drain and cool until easily handled, about 5 minutes. Chop finely.
Heat oil in a large skillet over medium heat; add onion. Cook and stir until translucent, about 5 minutes. Remove from heat; stir in chicken. Season with salt and black pepper.
Combine flour, hot milk, and egg yolk in a large bowl, stirring quickly with a fork until a soft and smooth dough forms.
Roll dough out on a lightly-floured surface. Cut into 1 1/2-inch squares. Place a teaspoon of chicken mixture in the center of each square. Fold the dough over the filling to form a triangle and pinch edges to seal. Bring the bottom corners together and pinch together.
Bring a large pot of lightly salted water to a boil. Cook dumplings in batches until they float to the surface of the water, about 10 minutes. Transfer to a serving plate with a slotted spoon. Pour melted butter over cooked dumplings so they do not stick together.