For once, water and oil work together to make a magical one-skillet weeknight chicken dinner. Potatoes cook alongside crispy chicken thighs — first they boil in water until tender, and then fry into golden nuggets in the remaining oil.
Sprinkle the chicken on both sides with salt and pepper. Melt the butter in a large skillet over medium-high heat. Add the chicken skin-side down and cook until golden brown and crispy and the skin releases easily from the skillet, 6 to 7 minutes.
Reduce the heat to medium and flip the chicken over. Scatter the potatoes, then lemon and then the garlic around the chicken. Add 1 cup water, red pepper flakes and 1/2 teaspoon salt. Cover and cook until the potatoes can be pierced easily with a fork, 8 to 10 minutes. Remove the lid and continue to cook until the liquid has evaporated, an instant read thermometer inserted into the thickest part of the chicken reaches 165ºF and the potatoes start to crisp from the remaining fat in the pan, 8 to 10 minutes. Sprinkle with parsley.
Divide the arugula among 4 plates and top with the chicken and potatoes. Spoon any juices left in the skillet over each dish.