Mince 2 to 3 garlic cloves until you have 1 teaspoon. Place in a medium bowl and add ½ cup plus 1 tablespoon sesame paste, ½ cup plus 1 tablespoon water, 3 tablespoons soy sauce, 3 tablespoons toasted sesame oil, 2 tablespoons creamy peanut butter, 2 tablespoons rice vinegar, 1 tablespoon packed brown sugar, and ¼ teaspoon kosher salt. Whisk until smooth.

Whisk 2 large eggs with a pinch of salt with a fork in a small bowl until well blended. Heat 1 ½ teaspoons vegetable oil in a large nonstick pan over medium heat until shimmering. Pour in the eggs in and tilt the pan to create a thin layer. Cook until the egg is set and crêpe-like with little to no browning, about 1 minute per side. Transfer to a cutting board and cut into thirds. Stack the portions and cut crosswise into ¼-inch-wide matchsticks. Transfer to a plate.

Bring a large pot of water to a boil over medium-high heat. Meanwhile, prepare the following, placing each on the plate of eggs as you complete it (you should get about ½ cup each): Cut ¼ English cucumber lengthwise into ¼-inch-thick planks, then stack a few planks and cut lengthwise again into ¼-inch-wide matchsticks; peel and cut 1 small carrot into thirds crosswise, then cut lengthwise into ⅛-inch-thick planks, and finally cut the planks into ⅛-inch-wide matchsticks; cut 2 ounces sliced ham into ¼-inch-wide matchsticks that are about 2 inches long.

Add 7 ounces dried Kuan Miao noodles to the boiling water and cook, stirring occasionally, until tender, about 4 minutes or according to package directions. Drain and rinse under cold running water, then drain well again. Transfer to a large bowl.

Add about ⅔ of the sauce to the noodles and toss to combine, adding 1 tablespoon water if needed to loosen the sauce. Add all the toppings and toss to combine. Taste and season with more sauce, kosher salt, or soy sauce as needed.