Heat 2 tablespoons extra-virgin olive oil in a large skillet over medium heat. Sauté chopped onion, garlic, and grated carrot until fragrant and translucent, about 3–4 minutes.

Add finely chopped mushrooms and toasted walnuts to the skillet. Cook, stirring occasionally, until mushrooms release moisture and turn golden brown, about 5–6 minutes.

Mix in tomato paste and cook for 1–2 minutes, stirring constantly, until it darkens to a rich red and clings to the mushrooms.

Add crushed tomatoes and vegetable broth, stirring to combine. Sprinkle in oregano, thyme, salt, pepper, and red pepper flakes. Bring to a simmer.

Reduce heat to low, cover partially, and let the sauce bubble gently for 15–20 minutes until thickened.

Stir well, then taste and adjust seasoning with more salt, pepper, or herbs.

Ladle the Bolognese over your favorite pasta or use in lasagna.