Make shallots and shallot oil: Scatter shallots evenly in an 8-in. square glass baking dish and pour oil on top. Microwave on high until shallots are deep golden brown, 9 to 12 minutes, stirring occasionally and watching closely toward end of cooking time to make sure they don’t burn. With a slotted spoon, transfer shallots to paper towels. Strain oil through a fine strainer. Set shallots and oil aside.
Prepare chicken: In a deep medium bowl, whisk together 3 tbsp. shallot oil with everything except chicken. Add chicken, turn to coat, and chill, covered, at least 1 hour.
Make dressing: Whisk 1/3 cup shallot oil with all ingredients in a bowl until combined. Set aside.
Heat a grill to medium-high (400° to 450°). Remove chicken from marinade and grill, covered, turning once, until cooked through and grill marks form, 10 to 15 minutes total. Transfer to a cutting board and let rest a few minutes.
Make salad: Toss lettuce, basil, mint, and cilantro in a large salad bowl. Cut chicken into small slices and toss gently with salad, half of melon, and about 1/3 cup dressing. Transfer to a platter. Top with peanuts and remaining melon, give salad a loose toss, and scatter with crispy shallots. Serve with more dressing on the side, and season to taste with salt if you like.
MAKE AHEAD: Shallots and shallot oil, marinated chicken, and dressing (recipe through step 3), chilled, up to 1 day; lettuces and herbs prepped and chilled in paper towel–lined resealable bags, up to 1 day