Heat a large skillet over medium heat.
Melt 1 Tsp. of butter to the pan and brown your sliced bread in the pan until golden brown.
Remove from heat and place them in a single layer on a baking sheet.
Spread a large spoonful of Béchamel sauce on top of each slice, then layer ham on top.
Sprinkle with Gruyere cheese and finely chopped fresh thyme.
Place sheet under the broiler on high for about 5 minutes or until golden brown and cheese is bubbling
Remove tray from oven when done and set aside.
Place butter to your skillet, on medium high heat.
Carefully crack each egg gently onto skillet leaving some room between edges of eggs. Season with salt and pepper and fry for about 2 minutes and edges of eggs are sizzling. Then carefully tilt pan towards you and spoon the butter and oil over eggs. Let cook for another minute.
When fried eggs are over easy with firm whites and yolk still runny, remove from heat.
Time to Plate.
Assembly:
Place one sandwich on each plate and layer one over-easy egg on top.
Garnish with grated fresh Parmesan cheese and a handful of micro greens if desired.
Serve with a simple Arugula side salad tossed with French balsamic vinaigrette, mandolin sliced radishes, and fresh grated Parmesan cheese or pommes frites.
