Take the chicken breasts, remove any fat. Place 1 chicken breast between pieces of plastic wrap and flatten with a meat mallet, until about 1/4-1/2 inch thick. Repeat for the remaining chicken.
Place chicken into a large baking dish. Add the shallots, 1/4 of the olive oil, and the juice of 2 lemons (not the seeds). Season with Pepper to taste.
Cover and marinate for 30 minutes, no longer, turning pieces of chicken after 15 minutes.
While chicken is marinating, preheat the grill to high and make the salad. Place the greens in a large bowl. Slice the red onion thinly, and add to greens. Slice the tomatoes into 1/2 inch slices, then in half again and add to greens. Cover for now.
Once chicken has marinated for 15 minutes, halve the potatoes and place in a pot. Add enough chicken broth (at least 1 cup, and water if needed) just to cover the potatoes. Squeeze the juice of 1 lemon into the pot, season with salt and pepper to taste.
Bring the potatoes to a boil, then reduce heat to medium. Cook the potatoes about 20 minutes, or until fork tender.
Grill the chicken on high about 5 minutes per side, or until just cooked through.
While chicken is grilling, make the vinaigrette. Put the Dijon mustard, vinegar, the juice of half a lemon, salt, pepper and garlic powder into a bowl. Whisk in the olive oil until emulsified. Set aside.
Drain potatoes when done, place back into pot. Drizzle a few tablespoons of olive oil onto the potatoes, salt, pepper, a tsp of lemon zest, juice of 1/2 a lemon, and the chopped basil. Mix to combine.
To serve, place salad onto a large platter in 1 corner. Place potatoes into another corner. Add the chicken pieces in the remaining space. Add the dressing on top of the salad, while spooning a small amount onto the chicken breasts as well. Serve with a few lemon wedges.