Working with one chicken breast half at a time, place it between 2 sheets of waxed paper and, using a meat pounder, pound to an even thickness of 3/4" thick. Season on both sides with salt and pepper. Place in a nonreactive dish large enough to hold them in a single layer. Add the lemon juice, olive oil and thyme. Turn the breasts over several times to coat. Cover loosely with plastic wrap and let marinate at room temperature for 1 hour or refrigerate overnight.
To make aioli, preheat a broiler (grill). Set the bell pepper on a baking sheet and place in the broiler about 6inches from the heat source. Broil (grill) turning with tongs, until skin is blistered and charred black on all sides, about 15 minute Enclose the pepper in a paper bag and let cool completely.
If broiling the chicken, leave the broiler on. If grilling the chicken, prepare a charcoal or gas grill for direct grilling over medium-high heat. Oil the grill rack.
Remove the stem from the pepper and discard. Slit the pepper open, then remove and discard seeds and ribs. Remove the blackened skin with a small knife or your fingers. Coarsely chop the pepper. In a blender or food processor, combine the roasted pepper, egg yolk, garlic, lemon juice, mustard, tomato paste, parsley, salt and pepper. Process until pureed. With the motor running, add the olive oil drop by drop. As the mixture begins to emulsify, add the oil in a slow, steady stream. Continue to process until the aioli is thick and has the consistency of mayonnaise. The aioli can be made a day ahead and stored in an airtight container in the refrigerator.
By Broiler: Remove chicken breasts from the marinade and arrange in a single layer on a rimmed baking sheet. Broil until lightly browned, 3-5 minutes. Turn and broil until lightly browned on the second sides and firm to the touch, 3-5 minute Transfer to a plate. Place the bread on the baking sheet and toast, turning once, until golden, 3-5 minutes total.
By Grill: Remove chicken breasts from the marinade and place over the hottest part of the fire or directly over the heat elements and grill until lightly browned, 3-5 minute Turn and grill until lightly browned on the second sides and firm to the touch, 3-5 minute Transfer to a plate. Place the bread slices on the grill rack and toast, turning once, until golden, about 2 min total.
Spread the bread slices with aioli. Place the chicken breasts on four of the slices. Put the remaining slices on top. Halve the sandwiches and serve at once.