Cook the pasta according to package directions. Reserve ¼ cup of the starchy pasta water, drain, rinse with cold water, and set aside.
Heat the oil in a sauté pan over medium-high heat. Once the oil is hot, add the vegan ground, season with the salt and pepper, and cook, stirring often, until heated through, 6 to 8 minutes. Remove from the heat and let cool at room temperature, 8 to 10 minutes.
Meanwhile, prepare the burger sauce. Place the vegan mayonnaise, ketchup, pickle relish, vinegar, mustard, and reserved pasta water sauce into a small bowl and whisk until thoroughly combined. Taste and adjust seasonings if necessary.
Place the pasta noodles in a large bowl and toss with ¾ of the sauce. Add the vegan ground, tomatoes, pickles, onions, and cheese, tossing again until combined. Place in the refrigerator to cool for at least 30 minutes, preferably 1 to 2 hours. When ready to serve, add the lettuce and remaining sauce, tossing to combine once more. Garnish with more vegan cheddar shreds, toasted sesame seeds and parley, and enjoy!