Line 4 four-inch ramekins with plastic wrap, leaving 4 inches of overhang. Cook the noodles in a large pot of boiling water, according to the package directions. Drain; rinse noodles under cold water. In a medium bowl, whisk the egg. Add the noodles; toss to coat. Divide noodle mixture among the ramekins. Fold the plastic over the noodles and press to compact. Refrigerate until set, 1 hour.
In a small bowl, whisk the mayonnaise, sriracha and sesame oil. In another bowl, combine the beef and the flavor packet. Form the beef into 2 four-inch patties.
In a large skillet, heat 1/2 tbsp. vegetable oil over medium-high. Add the patties and cook, turning once, about 8 minutes for medium. During the last minute of cooking, top each patty with a cheese slice and cover until the cheese melts. Transfer the patties to a plate. Wipe out the skillet. Add the remaining 1/4 cup vegetable oil; heat over medium-high. Unwrap the ramen buns and cook, turning once, until browned and crisp, 8 minutes. Transfer to paper towels.
Build burgers using ramen buns, patties, tomatoes, lettuce and sriracha mayo.