Heat oil in a large skillet over medium heat. Add onions and garlic to pan; cook 3 minutes or until fragrant, stirring often. Stir in 1/4 teaspoon salt, 1/4 teaspoon pepper, and tomatoes; bring to a boil. Reduce heat, and simmer 15 minutes, stirring occasionally.
Reduce heat to low. Working with one egg at a time, crack eggs over tomato mixture, about 1 inch apart in pan. Sprinkle eggs with remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Cover and cook 5 minutes or until desired degree of doneness. Remove from heat.
Arrange 4 wedges pane carasau on each of 4 plates; spoon 3/4 cup sauce over each serving. Top each serving with 1 egg, and sprinkle with 2 tablespoons grated cheese. Top each serving with 1 1/2 teaspoons chopped fresh basil.