1. In a small bowl, stir together garlic and mayonnaise. Cover and refrigerate to let flavors meld.
2. In a large skillet over medium heat, warm 1 Tbs. butter and the olive oil until melted and bubbly. Add onions and cook undisturbed for 8 minutes. Add the sugar and 1/4 tsp. salt, and continue cooking, stirring occasionally, until dark amber and caramelized, about 25 minutes. Transfer onions to a plate and reserve the pan. Chop 2 Tbs. of the prepared onions.
3. In the same pan over medium heat, melt the remaining 2 Tbs. butter. Add the garlic and chopped onions, and cook until fragrant, about 1 minute. Whisk in the flour until well blended. Slowly pour in the broth, whisking constantly until combined. Bring mixture to a simmer and let cook until slightly thickened, about 4 minutes. Stir in Worcestershire sauce, pepper, and remaining 1/4 tsp. salt.
4. While gravy cooks, toast rolls cut-side up in the oven, about 1 minute. To assemble each sandwich, spread garlic mayo on half a roll. Dip one fourth of roast beef and one fourth of onions in gravy, and place in the roll with cheese, if desired. Serve immediately with gravy for dipping and a green salad.