Bring 12 cups of water to a full boil and add 1 teaspoon of the salt. Gently, yet quickly, lower 6 eggs into the water. Boil the eggs for 4 minutes and 30 seconds.
Create an an ice bath while the eggs are boiling, using a large mixing bowl. Once the eggs have cooked for the allotted time, immediately transfer them the ice bath.
This next step is like one of those scenes from a Kung Fu movie where a budding warrior's patience and skill is tested by catching flies in chopsticks and what not. Seriously, find your moment of Zen because if you don't... you're gonna go all Hulk here. Gently tap the top and bottom of each egg on the kitchen counter. Then, while applying NO pressure other than what is needed to roll the egg with your palm, give it a few rolls. Now peel the eggs under a thin stream of warm water, carefully removing the membrane but being careful enough to not crack the egg white. Gently dry them with a paper towel and set aside. This is what they look like inside.
Warm the teaspoon of olive oil in a small frying pan and add in the diced red onion. Cook over a medium-high heat until the onions have caramelized and begun to just char a bit. Toss them with a pinch of salt and set aside.
In a large mixing bowl combine the 2 1/2 cups flour, teaspoon salt, 1/4 teaspoon ground pepper, tablespoon baking powder, and 1/2 teaspoon baking soda.
In a small saucepan, warm the 1/2 cup butter until just melted. Add in the 1 1/4 cup of buttermilk.
In a small mixing bowl, lightly whisk 2 eggs together and then mix in the butter-buttermilk mixture.
Pour the liquid mixture with the dry mix, stirring until the ingredients are mostly combined. Don't over mix! Fold in the caramelized onions, minced thyme leaves, and shredded cheese, setting some of the cheese aside for topping.
Pre-heat the oven to 400F and place the baking rack in the middle of the oven. Grease the inside of muffin tin, just six of them.
Place a heaping spoonful of the muffin batter into each greased space. Place an egg in the batter, pointy side up, wiggling it gently to set it. Then pipe the batter around each egg in a spiral pattern, completely covering the egg. Add more to the top so that if the batter slides at all, the egg is not exposed while cooking.
Cover the muffins with the remaining cheese and bake them in a preheated oven for 17 minutes at 400 degrees F, rotating halfway through the baking time. The crust should be firm and golden brown when done. Remove them from the oven and let cool for 10 minutes before serving.