Put the lemon juice, honey, and olive oil in a small bowl or jar. Season with salt and pepper and whisk to combine. Set aside.
Heat the olive oil in a large cast-iron skillet over medium-high heat. Season the peaches with salt and pepper. Add to the skillet and cook for 4 minutes or until soft and lightly browned. Remove from the pan and set aside. Add the onions to the pan and cook for 5 minutes or until soft, stirring occasionally. Remove from the heat and set aside.
Put the arugula, frisée, peaches, onion, basil, and almonds in a medium bowl. Add the dressing and gently toss to combine.
Arrange the salad on a large platter or on individual plates. Season with freshly ground black pepper, if desired. Serve with the goat cheese on the side.
Serve immediately.