In a medium-sized saucepan, melt the butter over medium heat. Add the chopped onion and minced garlic, cooking until they become translucent and fragrant.

Stir in the shredded chicken and season with salt and black pepper. Allow the flavors to meld together for a few minutes, then remove from heat and set aside.

In the same saucepan, bring the chicken broth to a boil. Reduce the heat to low and gradually whisk in the flour until it forms a smooth dough. Cook the dough for a few minutes until it pulls away from the sides of the pan.

Transfer the dough to a lightly floured surface and let it cool slightly. Knead the dough until it becomes smooth and pliable.

Take a small portion of the dough and flatten it in your palm. Add a spoonful of the shredded chicken mixture to the center of the dough. Carefully shape the dough around the filling to form a teardrop or drumstick shape. Repeat this process until you've used up all the dough and filling.

In a shallow bowl, whisk together the milk and beaten eggs. Dip each coxinha in the egg mixture, then roll it in breadcrumbs to coat evenly.

In a deep frying pan or pot, heat oil to around 350°F (175°C). Carefully place the coxinhas in the hot oil, frying them until they turn golden brown and crispy. Be sure to cook them in batches, so they have enough space to fry evenly.

Once cooked, remove the coxinhas from the oil and place them on a paper towel-lined plate to drain excess oil.