Prep the ingredients
In a 9x13-inch baking dish or other large, shallow dish, whisk together the marinade ingredients: olive oil, lemon juice, garlic, salt, dill, oregano, and pepper.
Cut each chicken breast in half across the middle and pound with a mallet if necessary to make each piece about an inch thick and a little bigger than a hamburger bun. Or, if using chicken tenders, see prep note below.
Place chicken pieces into the dish with the marinade and coat well., Leave to marinate at room temperature for 30 minutes or in the fridge for anywhere between 2 and 24 hours.
Cut bell peppers in half from stem end to tip and remove stems, ribs and seeds. Cut a slit of about an inch at both the top and bottom of each pepper half to help it relax into a flatter shape as it grills.
Cut halloumi into four slices, not super-thin or they might get too floppy on the grill.
Grilling Instructions
Prepare grill for direct heat.
Arrange chicken on grill with some marinade still clinging to it. Dredge pepper pieces through marinade and place on grill.
Grill for 5 minutes on the first side, then flip everything and grill chicken about 3 to 4 minutes more, until just cooked through. (Chicken tenders will cook much faster.)
Set chicken on a clean plate.
If peppers are soft enough for your liking, remove them too, otherwise leave them on while you grill the halloumi.
Dredge halloumi through marinade and place on grill. Cook 2 to 3 minutes on first side, until chewy and lightly grill-marked. Flip and cook one minute more.
Sauce & Serve
In a small bowl, stir together the sauce ingredients: sour cream, olive oil, lemon juice, shallot, parsley, dill, tarragon, salt, and pepper.
Assemble each sandwich by layering chicken, pepper, halloumi and arugula onto each bottom bun and spreading some sauce onto each top bun. Serve right away.