Preheat the oven to 400 degrees . In a large pot, combine the potatoes and salted water to cover by 1 inch. Bring to a boil and cook until fork-tender, 6 to 8 minutes. Drain and return the potatoes to the pot. Toss with 2 tbsp. butter, the parmesan and parsley; season with salt and pepper. Cover to keep warm.
Meanwhile, in a large skillet, heat 2 tbsp. EVOO over medium-high heat. Season the chicken and cook, turning once, until golden-brown, about 5 minutes. Transfer to a baking dish; roast until cooked through, about 10 minutes.
In the same skillet, heat the remaining 1 tbsp. each butter and EVOO over medium heat. Stir in the onions and vinegar and season with salt; cook, stirring occasionally, until the onions are caramelized, about 15 minutes. At the end, stir in 1/4 cup water and scrape up any browned bits.
Serve the chicken on a bed of spinach and top with the caramelized onions. Serve the potatoes on the side.