Put the bulgur, 2 teaspoons salt and 1 1/2 cups water in a saucepan and bring to a boil. Cover immediately, remove from the heat and let stand until all of the water has been absorbed, about 20 minutes.
Position an oven rack in the top of the oven, put a baking sheet on the rack and preheat the broiler on low. (If the broiler has just 1 setting, preheat the baking sheet on a rack 8 inches from the broiler.)
Add the lemon juice, mustard, garlic and shallot to a medium bowl and whisk to combine. Whisk in 1 teaspoon salt and a couple turns of ground black pepper. Drizzle in the olive oil, whisking until well combined. Pour half of the mixture into a large bowl or resealable plastic bag. Reserve the remaining half for the bulgur.
Add the chicken to the marinade, turning to coat evenly. (The chicken is ready for cooking immediately, but you can also marinate it, refrigerated, for up to 12 hours before cooking.)
Carefully remove the preheated baking sheet from the oven. Remove the chicken from the marinade (discard the marinade), sprinkle with salt and pepper and arrange skin-side up on the baking sheet. Broil until the chicken is crispy and golden brown and an instant-read thermometer inserted in the thickest part of each piece registers 175 degrees F, 15 to 20 minutes.
While the chicken is cooking, toss the bulgur with the tomatoes, parsley, mint, cucumbers and reserved marinade. Season with salt and pepper.
Transfer the chicken to a large platter and serve with the bulgur salad.