In a large bowl, combine the warm milk, sugar, and instant dry yeast. Stir gently and let sit for about 5-10 minutes until the mixture is frothy, indicating the yeast is active.
To the yeast mixture, add 1 cup of flour, salt, olive oil, and half of the melted butter (2 tbsp). Mix well until combined. Gradually add more flour until the dough comes together but is still slightly sticky. You may not need all the flour.
Turn the dough out onto a floured surface and knead for about 5-7 minutes, or until smooth and elastic. Add a little more flour if the dough is too sticky to handle.
Place the dough in a greased bowl, turning it to coat all sides with oil. Cover with a clean kitchen towel (tea towel) or cling wrap and let rise in a warm place for about 1 hour, or until doubled in size.
Punch down the risen dough and turn it out onto a lightly floured surface. Divide the dough into equal pieces (about 8-10 rolls). Flatten each portion slightly, and place a spoonful of cheese (or a black of cheese) and a sprinkle of Italian seasoning in the center. Gather the dough ball edges and pinch to seal, forming a ball.
Place the shaped rolls seam side down in a lightly greased air fryer basket, in a single layer, allowing some space between them for expansion. Cover and let rise for another 30 minutes.
Preheat your air fryer to 320 degrees F (160°C). Brush the tops of the rolls with the remaining melted butter. Set cook time for 8-10 minutes, or until they are golden brown and cooked through. You might need to cook in batches depending on the size of your air fryer.
Remove the rolls from the air fryer and let them cool slightly on a wire rack. Serve warm for the best flavor.