01 - In a small bowl, combine the warm milk, granulated sugar, and active dry yeast. Stir gently and let the mixture rest for 5 minutes until it becomes frothy and activated.
02 - In a large mixing bowl or stand mixer fitted with a dough hook, combine the all-purpose flour, kosher salt, softened butter, grated Parmesan, and minced garlic. Pour in the activated yeast mixture and knead until a smooth, elastic dough forms, approximately 8 minutes.
03 - Transfer the dough to a lightly oiled bowl, cover tightly with plastic wrap or a damp kitchen towel, and set in a warm, draft-free spot until the dough has doubled in volume, about 1 hour.
04 - Punch down the risen dough and divide it into 8 equal portions. Roll each into a smooth ball and arrange on a parchment-lined baking sheet, spacing them about 2 inches apart. Brush the tops generously with the beaten egg wash and sprinkle with additional Parmesan if desired. Bake at 375°F for 18 to 20 minutes until deeply golden. Allow to cool slightly on the pan.
05 - Pat the filet mignon medallions dry with paper towels and season both sides liberally with kosher salt and freshly cracked black pepper. Heat the olive oil in a heavy-bottomed skillet over medium-high heat until shimmering. Sear the medallions for 2 to 3 minutes per side for medium-rare, adjusting time for preferred doneness. Add 1 tablespoon of butter during the final minute of cooking and baste the meat with the melted butter. Transfer to a cutting board and rest for 5 minutes.
06 - In a small bowl, combine the softened butter, finely minced garlic, chopped parsley, and a pinch of salt. Mix thoroughly until evenly incorporated.
07 - Slice each warm roll in half horizontally. Spread a generous layer of garlic butter on both the bottom and top halves. Layer the bottom bun with arugula, a rested filet mignon medallion, a slice of provolone or Swiss cheese, and a spoonful of caramelized onions. Cap with the top bun and serve immediately while warm.