In a zip-top plastic bag, set inside a bowl combine lemon juice, honey, oil, garlic, 1 teaspoon salt, and pepper.
Place chicken breasts between 2 pieces of plastic wrap and pound with mallet until they are no more than ¾” thick. Transfer chicken to bag with marinade and toss to coat. Seal bag and marinate, refrigerated, at minimum 4 hours or overnight.
In a medium bowl combine bulgur, ¼ teaspoon salt, and 1 cup boiling water. Cover and set aside until bulgur is tender and has absorbed the water, about 10 minutes. If there is any excess water, drain bulgur well.
Prepare a medium-hot 2-zone grill fire, preheat a gas grill on medium-high heat for 10 minutes, or heat an indoor grill or grill pan on medium heat (for a grill pan, lightly brush with vegetable oil).
Remove chicken from marinade, allowing excess to drip off. Grill chicken, turning occasionally, until the internal temperature reaches 160°F, then transfer to a cutting board and let rest for 5 minutes.
While chicken rests, in a large bowl toss bulgur with arugula and olive oil and lemon juice to taste. Add grapes and toss again.
Divide bulgur mixture among 4 bowls. Thinly slice chicken and place on top of bulgur. Garnish with avocado.