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Preheat oven to 450 degrees. Line a rimmed baking sheet with parchment and spray with nonstick cooking spray.
Make béchamel: Add butter to a medium saucepan over medium-high heat. When foaming subsides, add flour and whisk to combine. Cook, whisking occasionally, until mixture bubbles and thins out a bit, about 1-2 minutes. Gradually add milk, whisking constantly. Bring mixture to a simmer and cook until thickened, about 2 minutes. Season to taste with salt, pepper, and nutmeg. Remove from heat and let cool slightly.
Place bread slices on baking sheet. Using a round biscuit cutter that fits within the perimeter of the slice, punch a hole in each piece of bread. Place punched bread along side of slices. Divide half of béchamel sauce among bread slices and holes, spreading to coat. Top bread slices with ham and cheese, being careful not to cover the hole too much!
Drizzle remaining béchamel over cheese and ham. Crack eggs into the center of each slice, season with salt and pepper, and transfer to oven. Bake until eggs are just cooked through and béchamel is bubbling, about 15 minutes. Remove from oven, top with chives, and serve immediately.