TJ Lees Taiwanese Sesame Cold Noodle

Name: TJ Lee’s Taiwanese Sesame Cold Noodle
Ingredients: 2 to 3 cloves garlic; 1/2 cup plus 1 tablespoon Chinese sesame paste; 1/2 cup plus 1 tablespoon water, plus more as needed; 3 tablespoons soy sauce, plus more as needed; 3 tablespoons toasted sesame oil; 2 tablespoons creamy peanut butter; 2 tablespoons rice vinegar; 1 tablespoon packed light or dark brown sugar; 1/4 teaspoon kosher salt, plus more as needed; 1 1/2 teaspoons vegetable oil; 2 large eggs; Pinch kosher salt; 1/4 medium English cucumber (about 3 ounces); 1 small carrot (about 2 1/2 ounces); 2 ounces thinly sliced deli ham (about 2 slices); 7 ounces dried Kuan Miao (Guan Miao) Taiwanese thin noodles