Make the Pesto: Combine Basil, Pistachios, Garlic, Parmigiano-Reggiano, Olive Oil, and Lemon Juice in a food processor. Process until smooth, adding more olive oil if needed to reach desired consistency. Season to taste.
Toast the Brioche Bread to a golden brown.
Whisk together the Eggs, Crème Fraîche, Sea Salt, and White Pepper in a bowl.
Cook the egg mixture in a non-stick skillet over low heat, stirring gently, until softly set and creamy.
Spread a generous layer of Pistachio Pesto onto each slice of toast.
Top with the creamy scrambled eggs.
Place a half Burrata on top of the eggs on each toast.
Sprinkle with Flaky Sea Salt and serve immediately.
