Heat 1 tablespoon of olive oil in a medium saucepan over medium-high heat.

Add half of the chopped onions and cook, stirring, until lightly golden, about 3 minutes.

Add half of the minced garlic and cook, stirring, for 1 to 2 minutes.

Stir in tomato puree, 1/2 teaspoon of salt, 1/2 teaspoon of pepper, and 3 1/2 cups of water.

Increase heat to high and bring to a boil, stirring.

Reduce heat to low and simmer, stirring occasionally, for 30 minutes.

Meanwhile, heat 1 tablespoon of olive oil in a large saucepan over high heat.

Add the remaining onions and cook, stirring, until lightly golden, about 3 minutes.

Add the remaining garlic and cook, stirring, for 1 to 2 minutes.

Add all the sliced mushrooms and cook, stirring often, for 3 minutes.

Add the remaining 1 tablespoon of olive oil and cook, stirring occasionally, until mushrooms are tender and begin to release their juices, about 3 to 4 minutes more.

Stir in the tomato mixture and bring to a boil.

Reduce heat to low and simmer, stirring occasionally, until flavors have blended, about 20 minutes.

Remove from heat.

Stir in the chopped basil, remaining 1/2 teaspoon of salt, and remaining 1/2 teaspoon of pepper.

Serve hot.