For the tomatoes:
Slice green tomatoes ¾ inch thick. Place in a shallow pan and pour buttermilk over the tomatoes. If you don't have buttermilk, you can make your own by adding a little lemon juice to milk, half and half or heavy cream. Cover and refrigerate for 15-30 minutes.
In another shallow pan mix the all-purpose flour and semolina flour with the salt and pepper. Stir with a fork to mix well.
Add canola oil ¾ inch deep in a non-stick frying pan. Heat over medium heat. Remove the tomato slices from the buttermilk and dredge in the flour mixture. Place the coated slices in the hot pan. You should hear it sizzle. The trick is to not let them get too brown on the outside before the inside gets cooked. Adjust the heat accordingly. Flip over and fry the other side until gold brown. Remove from pan and drain on paper towels. The tomatoes can be kept warm in a 250 degree oven until ready to use.
For the grits:
Bring 3 cups salted water in a saucepan to a simmer on medium heat. Slowly pour in 1 cup grits while whisking continuously. Reduce the heat to low and cook until the grits are done and the water absorbed. Follow directions on package if different from these directions. Remove the grits from the heat and add the grated cheese. Stir it in while the grits are still hot. Add melted butter and stir well.
Place the warm grits in shallow bowls, Add 1-2 tomato slices per serving and top each with a freshly poached egg. Season with salt and pepper as desired. Serve immediately.