Rinse the bulgur in a fine-mesh strainer under cold water for a few seconds, then drain well and set aside.
Melt the butter in a 3-quart / 2.8L saucepan over medium heat until it foams.
Add the broken vermicelli and stir constantly for 2 to 3 minutes until golden brown. Watch it closely, it can burn fast once it starts to color.
Add the chopped onion and cook for 4 to 5 minutes, stirring often, until soft and translucent.
Stir in the rinsed bulgur and coat it in the butter and onion for about 1 minute.
Pour in the stock, add the salt and pepper, and bring everything to a boil.
Lower the heat, cover the pan tightly, and simmer for 15 to 18 minutes without lifting the lid, until the liquid is absorbed and the bulgur is tender.
Remove the pan from the heat and let it sit covered for 10 minutes to finish steaming.
Fluff the pilaf with a fork, taste, and adjust the salt if needed. Serve warm.