Preheat the oven to 300 degrees, and put a rack in the center of the oven.
Whisk together the eggs, half-and-half (or water), and salt.
Coat a square cake pan with vegetable oil spray and pour the egg mixture into it.
Place the cake pan in a water bath.
Bake 35-40 minutes until the eggs are set, rotating the pan halfway through, then add thyme and pepper if desired.
Remove the pan from the oven and let it cool for 10 minutes. Run a knife around the edges of the pan, then tip the eggs out onto a cutting board. Cut the baked eggs into sandwich portions.
Assemble the sandwich with your choice of bread and toppings.
Notes:
Principles for Egg Sandwich Perfection
INGREDIENT SELECTION Anchor the flavor: Build the sandwich around a well-seasoned, savory ingredient such as ham, smoked salmon, kimchi, or salami. Add richness: Mayonnaise, cheese, Greek yogurt, or avocado contributes fat and creaminess. Layer in accents: Acidic, pungent, and or fresh choices such as herbs, tomatoes, onion, and jam work well.
ASSEMBLY Mitigate moisture: Excess juices don't just drip; they also lubricate the other ingredients, making them prone to slipping out with every bite. Pat wet ingredients dry and avoid super juicy ones in the first place. Don't stack slippery ingredients: Items such as multiple tomato slices will slide around and fall out of the sandwich. Toast bread lightly: Crunchy, heavily toasted bread will cause ingredients to slide out when bitten. Use creamy ingredients as glue: They will hold lighter ingredients such as chopped herbs in place.
MAKE AHEAD POTENTIAL Because the egg squares are relatively thin, they warm up quickly in the microwave and thus maintain their moisture and delicate texture. If you’d like to plan for a meal on the go, prepare the eggs in advance, cut them into squares, and let them cool completely. Stack the squares in an airtight container and refrigerate them for up to three days. To reheat, arrange the egg square(s) on a plate and microwave at 50 percent power until they’re warm (about 45 seconds for a single square or 2 to 3 minutes for four squares).
