Mix rosemary, thyme, olive oil, garlic, and venison in a zip top bag and refrigerate four hours.
Remove venison from refrigerator and pound each half to 3/4-inch thick. Season venison liberally with salt and pepper. Heat two tablespoons of olive oil in a medium-sized skillet over medium-high heat until skillet is almost smoking and oil is shimmering. Place venison in skillet for about four minutes on the first side, then turn over and cook for three to four minutes. Transfer to cooling rack and allow to rest.
Meanwhile, heat two tablespoons of oil in a medium-sized saucepan over medium-high heat. Add onions and garlic and stir for 30 seconds. Add tomatoes, tomato paste, basil, 1/2 teaspoon of salt, and 1/4 teaspoon of pepper. Bring mixture to a boil, then reduce it to a simmer for about 15 minutes.
Half each piece of toast on the bias and drizzle with olive oil. Place toast on the plates.
Slice venison against the grain and distribute equally over the toast. Spoon about 1/4 cup of the tomato mixture over the venison.
Heat two tablespoons of olive oil in a non-stick skillet over medium heat. Add three eggs at one time to the pan. Cover and cook at medium to medium-low heat for two–three minutes or until whites are set, but yolk is still soft. Gently remove the eggs from the skillet and place over the tomatoes. Repeat with remaining eggs.
Sprinkle one tablespoon of Parmesan Reggiano cheese over the egg. Garnish with parsley then serve immediately.