Bring broth, thyme, and bay leaves to boil, covered in a large saucepan. Meanwhile, heat oil in a large saucepan over medium heat until shimmering. Add onion and 1/2 tsp salt. Increase heat to medium-high and cook until lightly browned and softened (about 5 minutes).

Stir in garlic and cook 30 seconds. Stir in broth mixture, scraping up any browned bits. Add tortellini and cook until tender, 3-6 minutes.

Stir in spinach and cook until wilted, 2 minutes. Remove bay leaves and season to taste with salt and pepper.