Marinate the chicken. Slice the chicken breasts through the middle so each makes two thinner cutlets. Place in a gallon sized ziplock bag with the lemon juice and zest, garlic, olive oil, and salt and pepper. Seal and smoosh around so that the chicken is evenly coated with the marinade. Refrigerate for at least one hour or overnight.
Make the salad dressing. Place all ingredients in a jar (I use old jam jars for this). Seal and shake, shake, shake until combined. Refrigerate at least one hour. (Dressing will keep for 2 weeks in the refrigerator.)
Preheat a grill on high heat for 10 minutes. Grill the chicken for 3 minutes on the first side. Flip and continue cooking chicken until it's cooked through, about 3-4 minutes more. Set aside to cool. Slice into strips.
Place the arugula, quinoa, strawberries, sugar snap peas, and english peas in a large salad bowl. Drizzle over about half the dressing and toss to combine. Top with chicken, cheese and torn basil leaves. Drizzle over more dressing, to taste. Season with salt and pepper. Serve immediately.