Pass the canned tomatoes through a sieve to drain the excess liquid. Transfer to a frying pan, and mix with a hand blender. (Alternatively, blend in a blender and transfer to the frying pan.)
Add the remaining ingredients for the tomato sauce to Step 1, and evaporate the liquid. Please adjust the amount of sugar depending on the sourness of the tomatoes. You can keep this sauce in a jar.
Cut the mochi rice cake into pieces. Cut the onion, green pepper, and ham into thin strips.
Heat oil in the pan. Add the mochi, and coat with the oil. Add a small amount of water, cover with a lid, and cook the mochi over medium-low heat until softened. Be careful not to burn it.
When the mochi has melted, spread the sauce on top. Top with the vegetables and cheese. Cover with the lid. When the onion is cooked through, it's done.