1. Using a long, serrated knife, cut bread in half horizontally. Spread butter evenly over cut sides.
2. In a large bowl, beat eggs to blend, then add beef, onion, bread crumbs, broth, garlic, and oregano and mix well. Line a 12- by 15-inch baking sheet with waxed paper. Scrape meat mixture onto paper and pat into a 12-inch-wide round (or 1 in. wider than bread).
3. Prepare barbecue for direct heat.
If using charcoal briquets, cover firegrate with a single, solid layer of ignited coals; let them burn down to desired heat. Set grill in place and measure heat.
If using a gas barbecue, turn all burners to high and close lid for 10 minutes. Adjust burners to desired heat. Set grill in place and measure heat.
When grill is hot (you can hold your hand at grill level only 2 to 3 seconds), lay meat on barbecue: holding both ends of baking sheet, invert patty onto grill. Pull waxed paper off the patty and discard.
4. Cook patty until browned on bottom, about 7 minutes. To turn, use 2 rimless baking sheets, one as a pusher to slide patty onto the second sheet. When patty is on the baking sheet, invert first sheet onto it. Hold baking sheets together, turn over, and slide patty, browned side up, back onto grill. Continue to cook until well-done (no longer pink in the center, cut to test), 7 to 9 minutes.
5. During the last few minutes the meat cooks, if there is space, lay bread, cut side down, on grill to toast. If there is not enough room, toast bread after removing patty. Slide a baking sheet under the meat, then slide meat onto the bottom half of the toasted bread. Cut the top half of the bread into 8 wedges. Arrange bread wedges on burger. Cut burger and the bottom of the bread into wedges.