Add canola oil to a small skillet set over high heat; add tasso (or ham and cayenne) and cook until lightly browned. Drain and set aside to cool.

Preheat oven to 375º F. Generously butter interior of a 10-inch cast-iron skillet. Place in oven.

Whisk milk, corn, cornmeal, and a pinch of salt in a large microwave-safe bowl; microwave on high for 5 minutes. Stir and continue to microwave until thick, about 2 more minutes. Meanwhile, beat eggs in a bowl with a mixer until thick and airy, about 5 minutes.

Whisk ½ cup of cheese and ¼ cup of minced green onions into hot cornmeal mixture until thick. Whisk half of cornmeal mixture into eggs, then whisk in remaining cornmeal mixture until smooth. Fold in reserved tasso. Remove skillet from oven; swirl butter to coat sides. Pour batter into skillet; sprinkle with remaining ¼ cup cheese. Bake until puffed, golden, and set, 20 to 25 minutes.

Meanwhile, chop tomatoes and toss with olive oil, remaining green onions, salt, and pepper. Cut spoonbread into wedges; serve with tomato mixture.