Heat 1 tbsp olive oil in a large skillet over medium-low heat. Add onions, sprinkle with 1/4 teaspoon salt and 1/8 teaspoon pepper, cover, and cook 13-15 minutes, stirring occasionally, until softened.
Remove the lid, increase heat to medium-high, cook 3-5 minutes, stirring, until onions turn golden brown. Remove from heat.
Preheat grill to medium heat.
Season chicken fillet with salt and pepper. Place on grill rack, cover with lid, grill 15-18 minutes until charred stripes form on both sides and internal temperature reaches 74°C / 165°F. Transfer to plate and cover with foil to keep warm.
Lightly brush 4 baton slices with olive oil, place on grill, cover with lid, cook 1-2 minutes on one side until lightly charred. Remove from grill.
Cut the cooked chicken fillet into 1.5 cm thick slices.
On the charred side of each bread slice, layer chicken slices, caramelized onions, and Gruyère cheese. Place sandwiches on grill, cover with lid, cook 2 minutes until cheese melts. Top with chopped fresh parsley and serve immediately.