1. Season the ground beef with salt and pepper. From into 12 equal quarter-pound burgers.
2. In a large sauté pan, melt your butter over medium-high heat. Add the mushrooms and sauté until they become tender and start to caramelize.
3. Next, pour the Madeira into the mushrooms and cook together until the wine has reduced by half; turn off the heat and stir in the sour cream. Whisk everything together and season to taste with salt and (generously) with pepper.
4. Now, grill the burgers to your desired doneness. As they cook, halve the dinner rolls horizontally and place a large pinch of fresh arugula on the bottoms. Layer the done burgers onto the arugula. Just before serving, top each burger with a hearty spoonful of mushroom sauce and finish with the lid of the bun. Serve immediately.