Melt the butter, and combine with the milk and 1 beaten egg. Stir in the yeast. Let sit until the yeast begins to form bubbles, about 5-10 minutes.
In a large bowl, mix together the flour, sugar and salt. Once combined, add the milk mixture and mix until liquid is fully incorporated into the dough. Using your hands or a dough hook, knead the dough until it’s soft and springy - about 8 minutes (it should pass the windowpane test). If dough is too wet to knead, add more flour in small increments until it’s workable.
Place dough in a large bowl coated in oil, cover with a cloth, and let sit in a warm place until approximately doubled in size, about 1 ½ hours.
Make the pesto: combine all pesto ingredients in a food processor and process until combined.
Grease two 9-by-4-inch loaf pans. Divide the dough in half. Setting half aside for the second loaf, begin by rolling the first half out on a well floured surface to about a 10” by 10” square.
Spread half of the pesto evenly over the dough, leaving about a ½” border around all edges. Roll the dough with the filling away from you to form a tight roll. Repeat with the other half of the dough and the rest of the pesto.
Cut each roll of dough in half lengthwise (it’s okay it it’s not a perfect cut. Chilling the dough before cutting will make this step easier, but I went without chilling, and it ended up just fine).
For each set of cut rolls, lay one half (cut side up) vertically on your floured surface, and the other half (also cut side up) perpendicularly across it, intersecting in the middle. Cross the ends on either side of this intersection a couple of times each to make a twist. Transfer each twist to a prepared pan, and nestle it in as best you can. If any pieces fall off, just stick them in the corners of the pan. As it rises, the dough will fill in any little gaps.
Cover pans, and let dough rise for 60-90 minutes. Near the end of rising, preheat oven to 375 degrees.
Beat an egg in a small bowl, until very smooth (this will be your egg wash). Uncover pans and brush egg wash all over the tops of the risen loaves. Bake them at 375 until the tops are golden brown, about 25 to 30 minutes.
As soon as loaves come out of the oven, brush the top of each loaf evenly with 1 Tbsp olive oil and dust each with 1 Tbsp of parmesan. Let loaves cool slightly in their pans before removing them, slicing, and serving.