Heat oil in medium skillet. Add onion and garlic; sauté until softened, about 5 minutes. Set aside to cool while preparing remaining ingredients.
Mix eggs with thyme, salt, pepper, mustard, Worcestershire sauce, pepper sauce, and milk or yogurt. Add egg mixture to meat in large bowl along with crackers, parsley, and cooked onion and garlic; mix with fork until evenly blended and meat mixture does not stick to bowl. (If mixture sticks, add additional milk or yogurt, a couple tablespoons at a time until mix no longer sticks.).
Turn meat mixture onto work surface. With wet hands, pat mixture into approximately 9-by-5-inch loaf shape. Place on foil-lined (for easy cleanup) shallow baking pan.
Bake loaf until loaf registers 160 degrees, about 1 hour. Cool at least 20 minutes.