For the syrup: Combine syrup and cinnamon stick in a small saucepan. Bring to a simmer, and remove from the heat.Let syrup sit for 10 minutes to cool. Remove the cinnamon stick and return to heat. Bring the syrup to simmer again, and whisk in the guava. Serve warm.
For the toast: Preheat the oven to 275 degrees F. Whisk together the milk, sugar, cinnamon and eggs in a shallow bowl until smooth. Place the bread slices in a shallow dish and pour the egg mixture on top.While the bread soaks, heat 2 tablespoons of the butter and 2 tablespoons of the oil in a large saute pan or nonstick griddle over medium-high heat until the oil shimmers. Fry the bread in batches until golden brown on the bottom. Lower the heat slightly, flip the bread and cook until the bottoms are golden brown and the bread is cooked through as well. Keep warm in the oven while you make the remaining bread