Rinse the wild rice under cold water in a fine-mesh sieve and let it drain.
Heat olive oil or butter in a medium saucepan or Dutch oven over medium heat.
Add the chopped yellow onion, celery, and carrot; sauté, stirring occasionally, until the vegetables soften and the onion is translucent, about 5–7 minutes.
Stir in the drained wild rice and dried thyme; toast for 2 minutes, stirring constantly.
Pour in the broth, then add the salt and black pepper; stir to combine.
Bring to a boil over medium-high heat, then reduce heat to low, cover tightly, and simmer without lifting the lid for 45–50 minutes, or until the rice is tender and most of the liquid is absorbed.
Remove from heat and let sit, covered, for 10 minutes to steam and become fluffy.
Uncover and fluff the pilaf with a fork, taste, and adjust seasoning if needed. Serve warm.