For pesto, combine all ingredients in food processor and blend until smooth. Taste and adjust salt as necessary.
Spread 2 tablespoons of pesto on 4 slices of bread; set aside.
Slice Brie and divide evenly among remaining 4 slices of bread.
Put pesto-covered bread, pesto side down, on each Brie-covered slice.
Heat large skillet over medium heat and melt 2 tablespoons of butter. Add sandwiches to skillet.
Cook sandwiches until golden brown on one side, about 3—4 minutes.
When ready to flip sandwiches, transfer to plate. Add remaining 2 tablespoons butter to skillet.
When butter is melted, add sandwiches back to skillet with un-grilled side facing down. Cook for an additional 3 minutes, until cheese is fully melted and bread is crisp and golden brown on each side.
Remove from skillet, cut each sandwich in half, and serve immediately.
Recipe courtesy of Food52