Preheat the oven to 300 degrees.
Whisk together the eggs, half-and-half (or water), and salt.
Coat a round cake pan with vegetable oil spray and pour the egg mixture into it.
Place the cake pan in a water bath and cover with foil for the first half of the cooking time.
Bake until the eggs are set, then add thyme and pepper if desired.
Cut the baked eggs into portions for sandwiches.
Assemble the sandwich with bread and desired toppings.
