Heat the olive oil in a large skillet or Dutch oven over medium heat until it shimmers and begins to bubble softly.
Add the diced onion and carrots to the hot oil. Cook, stirring occasionally, until the vegetables are deeply golden and fragrant, about 15 minutes. The onions should be caramelized and the carrots softened.
Push the vegetables to one side of the pan. Add the sliced mushrooms to the empty side and let them sit undisturbed for 2-3 minutes to develop a crust. Then, stir and cook until browned and no longer spongy, about 8-10 minutes.
Stir in the tomato paste and cook for 2 minutes, allowing it to darken slightly and become fragrant, which intensifies its smoky flavor.
Add the minced garlic and cook for another minute until fragrant, ensuring it doesn’t burn.
Pour in the crushed tomatoes and vegetable broth. Add the fresh thyme sprig and bay leaf. Bring the mixture to a gentle simmer, then reduce the heat to low and let it simmer uncovered for 30-40 minutes, stirring occasionally, until the sauce thickens and deepens in flavor.
Remove the thyme sprig and bay leaf. Taste the sauce and season with salt and black pepper as needed. For extra smokiness, stir in a pinch of smoked paprika if desired.
Let the sauce rest for a few minutes off the heat to allow flavors to deepen. Serve over your favorite cooked pasta, garnished with chopped fresh basil if using.