Blanch the broccoli. Submerge the florets in boiling water for 3 minutes. Use a slotted spoon to transfer the florets to a bowl of ice water. Reserve one and a half cups of the hot liquid from the broccoli.
Add the broccoli water and the milk to the pan. Keep whisking until the mixture thickens a bit, about 3 minutes.
While the eggs are boiling, Whisk the sharp mustard into the sauce. Season to taste with the salt, freshly ground pepper and a pinch at a time of granulated white sugar until the flavor is balanced.
Arrange the potatoes and broccoli on the plate. Distribute the hard-boiled egg halves then drizzle the creamy mustard ham sauce over the top. Pour the remaining sauce into a pitcher and serve on the side.