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 frothy and bubbly, indicating the yeast is alive and active.
02 - In a large mixing bowl or stand mixer, combine the all-purpose flour, kosher salt, softened butter, grated Parmesan cheese, and minced garlic. Pour in the activated yeast mixture and knead until a smooth, elastic dough forms, approximately 8 minutes by hand or 5 minutes with a mixer.
03 - Cover the dough with a clean kitchen towel or plastic wrap and place in a warm, draft-free area. Allow the dough to rise until it has doubled in size, approximately 1 hour.
04 - Divide the risen dough into 8 equal portions and roll each into a smooth ball. Arrange on a parchment-lined baking sheet with even spacing. Brush each roll generously with beaten egg wash and sprinkle additional Parmesan on top if desired. Bake at 375°F for 18–20 minutes until the tops are deep golden brown. Allow to cool slightly on the pan.
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 shimmering. Sear the medallions for 2–3 minutes per side for medium-rare doneness. Add 1 tablespoon of butter during the final minute of cooking, basting the meat with the melted butter. Transfer to a plate and rest for 5 minutes.
06 - In a small bowl, combine the softened butter, minced garlic clove, chopped fresh parsley, and a pinch of salt. Mix thoroughly until all ingredients are evenly incorporated.
07 - Slice each cooled garlic Parmesan roll in half horizontally. Spread the garlic butter mixture generously on both the bottom and top halves. Layer baby arugula leaves on the bottom half, followed by a filet mignon medallion, a slice of provolone or Swiss cheese, and a spoonful of caramelized onions. Cap with the top half of the roll. Serve immediately while warm.