Rinse wild rice under cold water, then cook according to package instructions (approximately 45 minutes in 4 cups of water). Drain any excess water.

In a large pot, melt butter over medium heat. Add diced onion, carrots, and celery; sauté for about 5 minutes until softened.

Pour in vegetable broth and bring to a gentle boil. Reduce heat to low and let it simmer uncovered for about 15 minutes.

Remove thyme stems if using fresh thyme, then add the cooked wild rice and evaporated milk to the pot. Stir in salt and pepper to taste.

Continue to heat the soup on medium until warmed through, about 5 minutes. Adjust consistency with more broth or water if needed.

Ladle the soup into bowls and garnish with fresh herbs or black pepper.