Cut onions in half (from the root to stem end), and then each half into narrow wedges. Heat 1 1/2 tablespoons oil to shimmering in a 10 or 12-inch nonstick skillet or sauté pan set over medium high heat. Add the onion and sauté 8 to 10 minutes until golden. Season to taste with salt and pepper.  Remove about one-third of the onion from pan and keep warm.

Add drained tomatoes and all but about 1 tablespoon basil to pan and cook with remaining onions about 1 minute to heat.

Beat eggs until well combined in a large bowl with fork. Add onion-tomato-basil mixture to eggs, stirring just to mix through evenly. Heat the remaining oil in skillet. Pour egg mixture into hot skillet, stirring just to allow some curds to form, without making scrambled eggs.

Once the mixture has begun to come together in large curds, use a rubber spatula to spread the mixture evenly over the pan.  Cover and cook over low heat until fully set, about 15 minutes.

Top with reserved onions and basil; sprinkle with cheese. Let rest 2 or 3 minutes, then cut into wedges to serve.

*Canned tomatoes can be used to save time. If desired, fire-roasted canned tomatoes will add a nice smoky flavor to the frittata.