Prepare a grill for medium-high heat, about 375 degrees F. Alternatively, heat a grill pan over medium heat.
Meanwhile, place 1 chicken breast in a large resealable bag. Pound the chicken with the flat side of a meat mallet, starting in the center and working your way to the edges, until the chicken is 1/2 inch thick. Transfer to a baking sheet or large plate. Repeat with the remaining chicken.
Drizzle the chicken with 1 to 2 tablespoons grapeseed oil and then season each side with salt and pepper.
Grill the chicken until cooked through and grill marks appear, 3 to 5 minutes per side. About 2 minutes before the chicken is done, grill the halved lemons, cut-side down.
While the chicken is grilling, make the salad: Combine the arugula with the olives, tomatoes and feta in a large bowl. Tear the basil leaves and oregano and add them to the bowl. Add the juice from 1 of the grilled lemons, the olive oil and some salt and pepper. Toss together until evenly coated in with the dressing. Taste and add more salt or pepper, if needed.
Remove the chicken to a large platter. Mound the salad on top. Cut the remaining grilled lemon into wedges and place on the platter. Serve family style.