Thoroughly clean the beef shanks under running water (Hugasan mabuti ang buto-buto). Fill a large pot with cold water and add the cleaned beef shanks. Bring the water to a full boil (Pakuluan hanggang kumulo) - this will take about 10 minutes.
Once boiling, drain the water and rinse both the pot and the meat to remove impurities (Hugasan ang kaldero at karne). This first boiling helps achieve a clear broth later.
Return the cleaned meat to the pot and add fresh cold water until the meat is completely covered (Lagyan ng sapat na tubig). Add the quartered onions, whole peppercorns, and fish sauce (Idagdag ang sibuyas, paminta, at patis). Bring to a boil again, then immediately reduce heat to maintain a gentle simmer at 85-90°C/185-195°F (Ibaba ang apoy at pakuluan ng banayad).
Let the broth simmer gently for 2 to 2.5 hours (Pakuluan ng dahan-dahan ng dalawang oras), occasionally skimming any foam or impurities that rise to the surface (Alisin ang bumubula sa ibabaw). The meat should be tender but still holding its shape when pierced with a fork (Kapag malambot na ang karne pero hindi pa nagkakadurug-durog).
After the meat is tender, add the quartered potatoes and corn (Idagdag ang patatas at mais). Continue simmering for 10-15 minutes until the potatoes are almost cooked through (Pakuluan hanggang malambot ang patatas).
Add the celery pieces (Idagdag ang kintsay) and cook for another 3-5 minutes. The celery should retain some crunch (May crunch pa dapat ang kintsay).
Finally, add the pechay and green onions (Idagdag ang pechay at dahon ng sibuyas). Cook for just 2-3 minutes until the vegetables are bright green and crisp-tender (Hanggang lumambot ng kaunti ang gulay).
Taste the broth and adjust seasoning with salt as needed (Timplahan ng asin kung kinakailangan). The soup should be savory but not too salty, as diners can add more fish sauce to taste.
Serve hot in large, deep bowls (Ihain ng mainit sa malalaking mangkok). Place small dishes of fish sauce, calamansi, and chopped Thai chilies on the table so each person can season their portion to taste (Lagyan ng sawsawan: patis, kalamansi, at siling labuyo). Always serve with hot steamed rice (Ihain kasama ang mainit na kanin).
For the traditional way of eating bulalo, encourage diners to scoop out the bone marrow and add it to their rice or soup. The marrow can be seasoned with a little patis and calamansi for extra flavor (Masarap timplahan ng patis at kalamansi).