Heat a medium saucepan over medium heat, add the butter and allow to melt. Add the shallot and sauté, stirring frequently, until they are translucent and tender, about 5 minutes.

Add the flour and stir to combine, let cook about 1 minute, to cook the flour taste off. Add the thyme sprig to the pot then gradually add the milk and bring to a simmer, whisking constantly then reduce the heat to low and continue to cook until the sauce has slightly thickened, about 10 minutes.

Season with salt and pepper to taste and remove the thyme sprig or use a fine mesh strainer to strain the sauce. Return the sauce to low heat until ready to use to keep it warm.

Preheat broiler on high. Line a baking sheet with parchment paper.

Place the slices of bread on the baking sheet. Top 2 of the slices of bread with 3 slices of ham each. Top all of the slices with about 3 tablespoons of the béchamel sauce, then use a knife to evenly spread it over the bread.

Top each slice of bread with about 1/2 cup of the Gruyère cheese.

Broil, checking frequently, until the cheese is melted and bubbling, about 5 minutes.

Heat a medium nonstick skillet over medium heat. Add the canola oil and heat through. Gently break each egg into the pan and fry until the whites of the egg have set and the yolk has thickened but is still runny, about 5 minutes.

If the eggs start to sputter and bubble up, reduce the heat. Season with salt and pepper. When the eggs are ready use a spatula to remove the eggs from the pan onto a plate.

Place one of the cheese-topped slices on top of one of the pieces with the ham, cheese side up. Place one of the eggs on top. Repeat with the remaining sandwich. Serve immediately.