Heat grill or grill pan to medium heat.
Separate ground beef into one 4 ounce ball and a second smaller 2 ounce ball. Flatten the large ball into a 1/2-inch thick patty. Place feta cheese in the middle. Flatten smaller ball into a 1/4-inch thick patty and set it on top of the feta cheese. Pinch the two patties together to seal. Coat burger with 1/2 of the oil and season generously with salt and pepper.
Use the remaining oil to coat the ear of corn and season with salt and pepper.
Place corn and burger on the grill and cover with the vents open (if using a grill pan, don't worry about covering it.
Grill corn, turning frequently until lightly toasted on all sides, about 8-10 minutes, depending on the heat of your grill. Remove and set aside until cool enough to handle.
Grill burger, undisturbed until grill marks form on one side, about 8 minutes depending on the heat of your grill. Flip and continue cooking until done to your likeness, about 8-10 more minutes for medium. Remove and cover with foil to keep warm.
Cut kernels of corn off the cob with a sharp knife by standing the cob on one end and sliding the knife down each side. Place corn in a bowl and combine with scallion, jalapeño, mayonnaise, cilantro, lime juice, lime zest, and cayenne. Season with salt and pepper.
Toast flatbread on the grill or in a dry cast iron skillet.
Place burger on the flatbread, top with corn salad and arugula and dig in!
