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 fitted with a dough hook, combine the all-purpose flour, kosher salt, softened butter, grated Parmesan cheese, and minced garlic. Pour in the activated yeast mixture and knead for approximately 8 minutes until a smooth, elastic dough forms that pulls away cleanly from the sides of the bowl.
03 - Transfer the dough to a lightly oiled bowl, cover tightly with plastic wrap or a clean kitchen towel, and set in a warm, draft-free location. Allow the dough to rise until it has doubled in size, approximately 1 hour.
04 - Punch down the risen dough and divide it into 8 equal portions. Roll each portion into a smooth ball and arrange on a parchment-lined baking sheet, spacing them about 2 inches apart. Brush the tops generously with beaten egg wash and sprinkle with additional Parmesan if desired. Bake in a preheated oven at 375°F for 18 to 20 minutes until the rolls are deep golden brown. Transfer to a wire rack to cool slightly.
05 - Pat the filet mignon medallions dry with paper towels and season both sides generously with kosher salt and freshly ground 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 doneness. During the final minute of cooking, add the butter to the pan and baste the medallions with the melted butter. Remove from heat and let the meat rest for 5 minutes.
06 - In a small bowl, combine the softened unsalted butter, minced garlic clove, finely chopped fresh parsley, and a pinch of kosher salt. Mix thoroughly until all ingredients are evenly incorporated.
07 - Slice each warm garlic Parmesan roll in half horizontally. Spread the garlic butter mixture generously on both the bottom and top halves. Layer fresh arugula leaves on each bottom half, followed by a rested filet mignon medallion, a slice of provolone or Swiss cheese, and a spoonful of caramelized onions. Cap with the top half of the roll and serve immediately while warm.