In a small saucepan, heat 1/2 tsp olive oil on medium heat. Add onion and apple. Stir and cook until onion and apple are softened, about 5 minutes. Add remaining ingredients and stir to combine. Turn heat to low and simmer 10 minutes, stirring occasionally.
Heat 1/2 tsp oil in a large skillet over medium heat. Add ham and cook until warmed through. Remove ham and wipe out pan.
Butter one side of each slice of bread. Place 2 slices butter-side down in the warm skillet. Layer bread with ham, gouda, arugula and apple chutney. Spread mustard on unbuttered sides of remaining slices of bread. Place mustard side down on top of sandwiches in the pan. Cook until the bottom slice is golden.
Flip sandwiches and cook until second side is golden. Remove sandwiches from pan and serve hot.
*If you like things on the less-spicy side, reduce the red pepper flakes by half.