In a medium bowl, combine cream, oregano, and 3 tablespoons Cajun seasoning. Refrigerate for 6 hours or overnight. On 2 baking sheets, place bread slices in a single layer; let sit uncovered overnight.
Pour half of cream mixture into a shallow 13x9-inch baking dish; Whisk in eggs, flour, and 1 teaspoon pepper. Add bread, turning to coat with cream mixture. Set aside.
Pour remaining cream mixture into a small saucepan. Bring to a simmer over medium heat. In a small bowl, stir together cornstarch and 1 tablespoon cool water. Whisk cornstarch mixture into hot cream; continue cooking until slightly thickened. Stir in cheese and remaining 1 teaspoon pepper.
In a small bowl, combine shrimp, remaining 1 tablespoon Cajun seasoning, and 2 tablespoons olive oil, tossing to combine; set aside.
In a large skillet, add 2 tablespoons olive oil; add butter, and melt over medium high heat. Add bread in batches, and cook until brown on both sides, about 4 minutes per side.
Meanwhile, in a medium skillet, heat remaining 1 tablespoon olive oil over medium-high heat. Add shrimp, and cook until pink and firm, about 2 minutes per side.
To serve, layer 2 pieces French toast, fried egg, and shrimp onto individual serving plates; drizzle each serving with cream sauce. Garnish with fresh oregano, fresh baby arugula, and fresh avocado slices, if desired.