Thinly slice the scallions into a small bowl and toss with the olive oil and a small dash of salt.
In a bowl, season the ground beef with salt and pepper to taste. Pinch off a small amount and microwave to test for seasoning and adjust accordingly.
Toast your hamburger buns on a hot pan.
Separate the ground beef into 4 equal sections.
Break one of the sections into 2 parts, one part slightly larger than the other.
Press the slightly larger section into the bottom of a stuffed hamburger press until it is even and there is hamburger up the sides. Take care not to press so that the meat is too thin or the melted brie will leak from the middle of the burger.
Add the brie to the center, cutting it so that the cheese fits in the middle of the patty, not up the sides.
Spoon 2 small teaspoons of scallions in oil in the center of the patty, on top of the cheese.
Place the remaining, slightly smaller amount of the quarter section of ground beef over the top and press to seal the patty.
Remove the patty from the press and gently apply pressure to the vertical sides to finish sealing the top and bottom of the patty together.
Repeat with the remaining 3 sections of ground beef.
Cook the patties on a grill pre-heated to medium high.
Place the patties gently on the grates and allow them to sit untouched until flipping. Cook to desired doneness (5 minutes per side for a burger cooked to medium), keeping in mind that these are thicker and might take longer.
Dress to taste and serve.