Heat 2 tablespoons olive oil and 1 tablespoon butter in a large skillet over medium heat; add sliced onions, 1 teaspoon sugar, and a pinch of salt and cook, stirring occasionally, until deep golden brown, 20–25 minutes. Transfer onions to a bowl when done.

Increase heat to medium-high, add remaining 1 tablespoon olive oil and brown the cut chicken in a single layer for 3–4 minutes until golden; season lightly with salt and pepper and transfer to the bowl with onions.

Reduce heat to medium, add remaining 1 tablespoon butter, then add orzo and toast for 1–2 minutes; stir in minced garlic for 30 seconds and sprinkle flour over the orzo, stirring another 30 seconds.

Deglaze with 1/2 cup additional hot stock plus 1 tablespoon white vinegar (or 1/2 cup hot stock alone), then add 2 cups hot stock and 1 cup hot stock reserved, Dijon, Worcestershire sauce, and pepper; bring to a gentle simmer and cook uncovered for 3–4 minutes until orzo has absorbed some liquid but is undercooked.

Return browned chicken and caramelized onions to the skillet and stir to combine. If using an ovenproof skillet, top with shredded cheddar; otherwise transfer mixture to a greased 9×13-inch baking dish and sprinkle cheese over the top.

Cover tightly with foil and bake at 375°F for 18 minutes, then remove foil and bake 8–10 more minutes until cheese is bubbly and chicken reaches 165°F. Let rest uncovered for 5 minutes before serving and garnish with chopped parsley.