Place egg, Pernod, if using, garlic, mustard, and lemon juice in the bowl of a food processor; season with salt and pepper. With the motor running, slowly add oil and process until an emulsion is formed. Add buttermilk, green onions, and 1/2 cup cheese; pulse to combine. Keep refrigerated until ready to use, up to 1 day.
Place greens in a large bowl; sprinkle with cheese and season with salt and pepper. Add enough dressing to just coat; toss to combine. Divide greens evenly between 4 plates; top with fried oysters. Sprinkle with cheese and drizzle with more dressing, if desired. Serve immediately.