Place stale bread in a shallow dish just large enough to tightly fit the slices in one layer.

Whisk eggs, salt, sugar, vanilla, cinnamon, cayenne, and heavy cream for batter together in a bowl and pour over bread slices. Use tongs to toss to coat evenly. Spoon excess batter over bread slices if needed.

Wrap and refrigerate until bread has absorbed the batter or is fully saturated (2 to 12 hours).

To make honey syrup, combine honey, water, chili flakes, and salt in a small saucepan. Simmer for 30 seconds and set aside.

Remove bread from refrigerator and press sliced feta into the top of each piece (about 2 tablespoons per slice).

Heat butter and olive oil in a nonstick skillet over medium-high heat. Place bread cheese-side down. Cook until caramelized and browned (~4 minutes). Turn over, reduce heat to medium, and cook other side until golden brown (~3 minutes).