01 - Combine warm milk, granulated 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 all-purpose flour, kosher salt, softened butter, grated Parmesan, and minced garlic. Pour in the activated yeast mixture and knead for approximately 8 minutes until a smooth, elastic dough forms that springs back when pressed.
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. Allow the dough to rise until doubled in volume, about 1 hour.
04 - Punch down the risen dough and divide it into 8 equal portions. Roll each into a tight ball and arrange on a parchment-lined baking sheet, spacing about 2 inches apart. Brush the tops generously with beaten egg and sprinkle with extra Parmesan if desired. Bake at 375°F for 18 to 20 minutes until deeply golden. Cool on the pan for 5 minutes before handling.
05 - Pat the filet mignon medallions dry with paper towels and season both sides liberally with kosher salt and freshly cracked black pepper. Heat olive oil in a heavy-bottomed skillet over medium-high heat until the oil just begins to shimmer. 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 butter to the skillet and spoon the melting butter over each medallion to baste. Transfer the seared filets to a cutting board and rest for 5 minutes to allow the juices to redistribute.
07 - In a small bowl, combine the softened butter, finely minced garlic, chopped parsley, and a pinch of salt. Mix thoroughly until all ingredients are evenly incorporated and the spread is smooth.
08 - Slice each warm roll in half horizontally. Spread the garlic butter generously on both the cut bottom and top halves. Layer arugula leaves on the bottom, followed by a filet mignon medallion, a slice of cheese, and a spoonful of caramelized onions. Cap with the top half of the roll and serve immediately while warm.