Heat the olive oil in a pot over medium heat. Once hot, add the onions, carrot, garlic, celery, and mushrooms. Cook, stirring occasionally until the mushrooms soften. Season with salt and pepper. Stir in the flour and cook for 1 minute more.
Stir in the broth and thyme, then bring to a simmer for about 20 minutes.
Taste and adjust the seasonings as needed, before serving. Serve with crackers.