Add butter to a cold saucepan, along with olive oil, crushed Aleppo pepper, salt and pepper. Turn heat on low, melt butter and bloom the Aleppo pepper. Hold warm for service.
To prepare the yogurt base:
In a mixing bowl, add yogurt, pepper, fennel seeds, garlic, shallot, tarragon, lemon zest and juice. Drizzle in olive oil, whisking until smooth and fully combined. Add salt to taste. Hold at room temp or cold for service.
To serve:
In individual bowls or ramekins, distribute the spiced yogurt and top with a poached egg. Generously drizzle warm Aleppo Pepper sauce over the top, and garnish with chives. Serve with Barbari bread.