Preheat the oven to 180°C (350°F) and set the oven to the broil/grill setting when ready to toast the bread.
In a medium bowl, combine the softened butter, chopped sun-dried tomatoes, fresh basil, sun-dried tomato oil, and Parmesan cheese. Mix until everything is evenly incorporated and a thick, spreadable mixture forms.
Slice the brioche or ciabatta into thick pieces, or slice the loaf lengthwise if you prefer to serve it as one large cheese bread. Place the bread on a baking tray lined with parchment paper or foil.
Spread the sun-dried tomato and basil butter generously over the surface of the bread. Make sure the mixture reaches close to the edges so every bite is flavorful.
Sprinkle the mozzarella or provolone evenly over the buttered bread.
Add the Parmesan on top, followed by a few extra chopped sun-dried tomatoes if desired.
Place the bread under the broiler and cook until the cheese is fully melted, bubbling, and golden brown in spots. Watch carefully: the bread can brown quickly under the broiler.
Remove the bread from the oven and allow it to cool for 1–2 minutes. Finish with freshly chopped basil and serve warm.