Brush the Brie with some of the oil. Grill until softened and grill marks appear, 3 to 4 minutes per side. Carefully transfer to a serving platter or board.
Toss the strawberries in the brown sugar in a bowl until coated. Brush the grill with the remaining oil and add the strawberries cut-side down. Grill until the sugar is caramelized and the strawberries are tender, 1 to 2 minutes per side.
Arrange the strawberries around and on top of the Brie. Garnish with the thyme and sprinkle with the fleur de sel and red pepper flakes. Serve warm with crackers.
Try a drizzle of balsamic glaze for a decadent touch.