Heat a grill or grill pan to high. Char the chiles, turning as needed, until blackened on all sides, about 10 minutes. Place in a large bowl and cover with plastic wrap. The steam will help the peels come off more easily. Let cool for about 10 minutes. Once the peppers are cool enough to handle, use your fingers to rub off the charred peel. I like to do this over the sink. Make a slice into the side of the chile and cut away and discard the stem, seeds, and veins. Finely chop the chiles. Put in a medium bowl and season with ¼ tsp. salt.
Mix the mayonnaise with the garlic in another medium bowl.
Season the ground beef with remaining ½ tsp. salt in a large bowl and combine. Divide into 4 portions and form ½ inch thick patties. Use your thumb to create a dimple in the center of each burger. This helps avoid “burger bulge”.
Preheat your grill or griddle to medium (350° to 450°). Grill the burgers, turning once and sprinkling with about ½ tsp. tamari on each burger side. The burgers should cook in approximately 7-8 minutes, for medium. During the last minute, lay a slice of cheese on each burger, to melt it.
Meanwhile, brush melted butter on the cut sides of the rolls and toast on grill or griddle, about 2 minutes.
Spread garlic mayonnaise on each side of the rolls. Add burgers and top with chile mixture.
When forming your burger patties, make a dimple in the middle. This prevents your burger from bulging up in the middle and keeps it nice and flat.