First, prepare pot for eggs: Fill a shallow saucepan with 2-3 inches water and bring to a simmer.
Then, prepare the béchamel: In a medium saucepan over medium heat, melt the Butter (1 tablespoon).
Add the Onion (2 tablespoon) and the Kosher Salt (to taste) and cook about 5 to 7 minutes or until the onion is soft but has not begun to color.
Turn the heat to very low, add the All-Purpose Flour (1 tablespoon) and stir to combine it with the onion and butter.
Continue to cook over low heat until the flour is absorbed, stirring constantly so that it doesn't brown, about 2 minutes or so. Slowly stir in the Milk (1 cup). Drop in the Bay Leaf (1).
Over medium to medium-high heat, bring the mixture to a boil then reduce the heat to its lowest setting and cook for about 15 minutes, stirring occasionally to prevent the sauce from burning on the bottom of the pan.
Taste and cook longer if the taste of raw flour is still detectable. The mixture should be thick, but if it's too thick and becoming difficult to stir, you'll need to whisk in a little more milk. Remove the bay leaf and discard.
Meanwhile, preheat the broiler. Place the slices of Bread (2 slice) on a rack on a sheet pan (or a broiling pan) and broil them about a minute on each side. Remove pan from the oven.
Spread about 1 tablespoon of béchamel over each slice of bread. Top with Black Forest Ham (4 slice). Top with Gruyère Cheese (to taste). Set aside.
Crack Egg (2) into a small bowl or ramekin. Add Distilled White Vinegar (to taste) into the pot of simmering shallow water. Adjust the heat so that the water is barely simmering — get the water to a simmer, then turn it down so you don't see any bubbles.
Use the handle of a wooden spoon to make a whirlpool in the water, then drop one egg into the center of the whirlpool. Repeat with the other egg. Adjust the heat to keep the water just below a simmer. Set the timer for 3 minutes.
When the eggs have cooked for 3 minutes, place the toasts under the broiler and cook until the cheese is bubbling and starting to brown. Remove from the oven. Sprinkle with the Fresh Thyme Leaves (to taste).
Meanwhile, using a slotted spoon, lift one egg up from the water and shake it. The yolk should jiggle a little bit, but shouldn't look too loose. When the eggs look cooked to your liking, remove them with a slotted spoon and transfer to a paper towel-lined plate.
Top each sandwich with a poached egg. Sprinkle with a pinch of Kosher Salt (to taste) and fresh Freshly Ground Black Pepper (to taste).