Heat the olive oil in a skillet. Cook the onion until soft, add the garlic, and cook for 30 seconds. Stir in the shredded chicken, cream cheese, parsley, and a pinch of salt.
Cook until the mixture is cohesive and no longer wet. Transfer to a bowl and cool completely.
Make the dough:
Bring the chicken stock, butter, and salt to a simmer in a large pot. Add the flour all at once and stir vigorously until a smooth dough forms and pulls away from the sides.
Turn the dough onto a lightly greased surface and knead carefully until smooth. Cool until warm but comfortable to handle.
Shape and fry:
Divide the dough into 18 pieces. Flatten each piece, place a spoonful of filling in the center, and pinch the dough around it into a sealed teardrop shape.
Dip each coxinha in beaten egg, roll in breadcrumbs, and fry at 350 F (175 C) until deeply golden. Drain on a rack and serve hot.
Notes
Cool the filling fully before shaping so the dough does not soften.
Keep the oil near 350 F (175 C) for an even crust.
Use a wire rack rather than paper towels to preserve crispness.