Fit your food processor with a steel blade. Place the bread flour, yeast, sugar, salt, Italian seasoning, garlic, 1/4 cup grated Parmesan and softened butter in the processor; pulse to blend. Stir together hot water, milk and egg. Start the processor and slowly pour in the 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 the 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 10x12 inch rectangle. Use a knife or pizza cutter to slice the dough lengthwise into ten strips, and then cut these strips in half. Brush tops of strips with 2 tablespoons melted butter, and sprinkle with 1/4 cup Parmesan cheese. Cover, and let rise until nearly double, about 30 minutes. Meanwhile, preheat oven to 375 degrees F (190 degrees C).
Bake in preheated oven for 18 to 23 minutes, or until golden brown. Turn broiler on, and brush breadsticks with remaining melted butter and Parmesan cheese. Broil just until the cheese starts to turn light brown, about 2 to 3 minutes. Serve warm.