Fit your food processor with a steel blade. Place flour, yeast, sugar, salt, Italian seasoning, garlic, 4 tablespoons Parmesan cheese and 15g butter in the processor; pulse to blend. Whisk together warm water, milk and egg. Start the processor and slowly pour in milk mixture into the feed tube until a ball forms around the blade. Run for 30 seconds to knead. Remove dough from processor and knead briefly by hand.
Lightly oil a large bowl, place dough in the bowl and turn to coat with oil. Cover with a damp cloth and let rise in a warm place until doubled in volume, about 1 hour.
Deflate the dough and turn it out onto a lightly floured surface. Roll the dough into a 25x30cm rectangle. Use a knife or pizza cutter to slice the dough lengthways into 10 strips, and then cut these strips in half. Brush tops of strips with 30g melted butter and sprinkle with 20g Parmesan cheese. Cover and let rise until nearly double, about 30 minutes. Meanwhile, preheat the oven to 190 C / Gas mark 5.
Bake in the preheated oven until golden brown, 18 to 23 minutes. Turn grill on and brush breadsticks with remaining melted butter and Parmesan cheese. Grill just until cheese starts to turn light brown, 2 to 3 minutes. Serve warm.