01 - Combine warm milk, sugar, and active dry yeast in a small bowl. Stir gently and let stand for 5 minutes until the mixture becomes foamy and bubbly, indicating the yeast is active.
02 - In a large mixing bowl or stand mixer, combine the flour, 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 - Cover the dough with a clean kitchen towel and place in a warm, draft-free area. Allow the dough to rise until doubled in size, approximately 1 hour.
04 - Punch down the dough and divide it into 8 equal portions. Shape each into a round ball and arrange on a parchment-lined baking sheet. Brush the tops generously with beaten egg wash and sprinkle with additional Parmesan if desired. Bake at 375°F for 18 to 20 minutes until golden brown. Cool slightly on the pan.
05 - Season the filet mignon medallions generously on both sides with kosher salt and freshly ground black pepper. Heat 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.
06 - During the final minute of cooking, add 1 tablespoon of unsalted butter to the skillet and baste the medallions with the melted butter. Transfer the meat to a cutting board and rest for 5 minutes to allow the juices to redistribute.
07 - In a small bowl, combine the softened butter, minced garlic, chopped parsley, and a pinch of salt. Mix thoroughly until all ingredients are evenly incorporated.
08 - Slice each roll in half horizontally. Spread the garlic butter mixture generously on both the bottom and top halves. Layer arugula leaves, a rested filet mignon medallion, a slice of cheese, and caramelized onions on the bottom half. Cap with the top half of the roll. Serve immediately while warm.