In a shallow dish or container, combine eggs, half and half, brown sugar, cinnamon, nutmeg, and salt. Whip together ingredients with a whisk or fork until eggs are completely scrambled and ingredients are well combined.
Preheat a griddle pan or electric griddle over medium heat. Once the pan is preheated, add the butter to the pan and allow it to melt completely, and coat the entire bottom of the pan.
Dip each slice of the bread into the egg mixture and then flip it over so that the mixture soaks into both sides of the bread. Remove the bread from the egg mixture, allowing the excess egg to drip off from the bread, and then place it directly on the buttered pan. Let the french toast cook for 1-2 minutes on each side until golden brown. Repeat with the remaining slices of bread.