If using rice noodles, submerge them in warm water for 30 minutes until softened but still slightly firm. Drain and set aside. For fresh wide wheat noodles, cook them in salted boiling water for 4-5 minutes, stirring once. Drain and rinse thoroughly under cold running water.
In a small bowl, whisk together the vegetarian oyster sauce, tamari, sugar, tamarind concentrate, and water.
Heat olive oil in a large pan or wok over medium-high heat. After 90 seconds, when the oil is hot, add the shallots and cook for 3-4 minutes, stirring frequently, until lightly golden.
Add the garlic and bird’s eye chilies, stirring for 30 seconds until fragrant.
Add the carrots and pickled green peppercorns (if using), and the Chinese broccoli. Sauté for another 3-4 minutes until the stems are just tender.
Increase to high heat and add the drained noodles. Pour in the prepared stir-fry sauce and toss everything together, stirring continuously for 2-3 minutes, until the noodles are evenly coated and heated through. Stir in the Thai basil leaves and cook for 30-60 seconds until just wilted. Remove from heat.
To make fried basil for garnish, heat 1 inch (2.5 cm) of oil in a small pan over high heat. After 2 minutes, when the oil is hot, carefully add the Thai basil leaves and fry for just 5–10 seconds until crisp. Use a slotted spoon to transfer them to a kitchen towel-lined plate.
Serve the noodles hot, garnished with additional bird’s eye chilies, cilantro leaves, and fried basil, if desired.