Make the glaze: Bring the port to a simmer in a small saucepan over medium-high heat. Cook until slightly reduced and syrupy, about 10 minutes. Remove from the heat and whisk in the truffle oil, truffle salt and sugar; add more truffle oil to taste. Let cool slightly.
Make the aioli: Whisk the mayonnaise, truffle oil, garlic and lemon juice in a small bowl; add more truffle oil to taste. Set aside.
Cook the burgers: Preheat a cast-iron or other heavy-bottomed skillet over high heat. Lightly brush the skillet with vegetable oil. Form the beef into four 3/4-inch-thick patties, 4 inches each. Cook until the bottoms are seared, 4 to 5 minutes. Flip and cook about 4 more minutes for medium. Top each burger with a slice of cheese and let melt, about 1 more minute; remove to a plate. Brush the cut sides of the buns with melted butter and toast in the skillet, 30 seconds to 1 minute.
Spread the aioli on the bottom buns; top with the burgers. Whisk the glaze and drizzle over the burgers; cover with the top buns.
Photograph by Steve Giralt
Recipe courtesy of Umami Burger for Food Network Magazine