Add the strawberries to a rimmed baking sheet. Drizzle with the olive oil, honey and balsamic vinegar, then sprinkle with the salt. Toss to combine and spread in an even layer. Bake for about 20 to 25 minutes, stirring once halfway through, until tender and the berries have released most of their juices. Let cool for about 5 to 10 minutes.
For the sandwiches:
Heat a large griddle or a large nonstick skillet over medium heat.
Set 4 slices of the bread on a work surface. Top each with some of the sliced brie, the roasted strawberries (using a slotted spoon to get rid of excess juices), the basil, arugula and finally another piece of bread. Spread some of the butter on the top and bottom of each sandwich.
Transfer the assembled sandwiches to the heated griddle and cook, carefully flipping once halfway through, until golden brown and the cheese is melted. Serve immediately.