Spread each of 4 slices of bread with 1/2 tablespoon of butter. On the opposite side spread each with 1 tablespoon of the apple butter. Top each slice of bread with a couple of slices of turkey, cover that with 2 to 3 slices of brie (enough to cover the turkey), and finally, arrange 2 to 3 arugula leaves on top. Spread the remaining 4 bread slices with 1/2 tablespoon of butter and lay them, buttered sides down, over the cheese press gently.
Heat a large cast iron or other heavy skillet over medium-low heat. Add the remaining 1 tablespoon butter and let it melt to evenly coat the bottom of the pan. Add as many sandwiches as will comfortably fit (either 2 or 4) and cook 2 to 3 minutes until the bread is golden brown and the cheese begins to melt. Flip and cook 2 to 3 more minutes to brown the bread. (Add another tablespoon of butter and cook the remaining sandwiches, if you need to.) Cut the sandwiches in half and serve hot.