Set a rack in the middle of the oven and preheat to 375 degrees. Coat the inside of a baking dish with one teaspoon of the softened butter. Use the rest to spread on both sides of the bread slices.
Heat a large skillet over medium-high heat and toast the bread until golden brown, 1 or 2 minutes per side.
In a small saucepan over low heat, bring the cream and garlic cloves to a simmer until fragrant with garlic. Toss the garlic. Add salt, pepper, and cayenne (if using) to taste.
Lay the toasted bread in a baking dish and spoon 2 to 3 tablespoons of the hot cream over each bread slice. Crack eggs into a small bowl, then gently pour atop slices of bread—take care not to break the yolks. Slowly pour the rest of the cream over the eggs and immediately transfer the baking dish to the oven.
Bake until the whites are set and the yolks are cooked to your preference. (Runny yolks will take around 13 or 14 minutes.) Remove from oven and top with the bacon and chives. Add grated cheese if using and serve immediately.