Place the chicken breasts between two sheets of parchment or plastic wrap. Pound to an even 1/4-inch thickness using a meat mallet or rolling pin. Season both sides with salt and pepper. Set aside.
In a large mixing bowl, combine arugula, tomatoes, fennel, red onion, and cucumber. You can eyeball it and drizzle lemon juice and olive oil straight into the bowl like I usually do—or make the dressing (below) like a civilized human.
For civilized humans: in a small bowl, whisk together the lemon juice, olive oil, salt, and pepper. Toss it with the salad just before serving.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat, or preheat a grill to 475 F and brush the chicken with olive oil. Cook chicken for 4–5 minutes per side until golden and cooked through. Cook the asparagus in the same pan or grill for 6-7 minutes, until tender, but still have a bite.
Place each chicken breast on a plate and pile the salad right on top. Plate asparagus on the side. Drizzle with any extra dressing or pan juices. Serve immediately.