1. Heat oil in a heavy pot, 10 to 12 inches in diameter. Add onion and ham/bacon; saute, stirring occasionally, until onions soften, 3 to 5 minutes. Add tomatoes, cover and cook over medium heat until tomatoes start to look like a thin sauce, about 10 minutes.
2. Stir in rice and 1 teaspoon salt or to taste. Add 3 cups of the broth/water mixture and bring to a boil, stirring occasionally. Reduce heat to simmer and cook, stirring occasionally, until pan bottom is dry when rice is pulled back with spoon, 8 to 10 minutes.
3. Add wine, stirring frequently until absorbed. Then add 1/2 cup of the broth/water mixture at a time, stirring constantly until each addition is absorbed; cook until rice is creamy but still somewhat firm in center (add more water in 1/2 cup increments if broth/water mixture runs out), 10 to 12 minutes longer.
4. Stir in cheese. Serve on a wide platter or individual plates, garnish with basil, and pass additional cheese separately.
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