Cook the Potatoes: Place the chopped potatoes in a pot and cover with cold water.
Bring to a boil over high heat, then simmer for about 10 minutes until tender.
Drain well and return the potatoes to the hot pot, off the heat.
Mash and Season: Using a masher, mash the potatoes until smooth.
Stir in the Greek yogurt, vegetable broth, Dijon mustard, nutmeg, sea salt, and white pepper until well combined.
Cover to keep warm and set aside.
Steam the Broccoli: While the potatoes cook, add an inch of water to a separate pot fitted with a steamer basket and bring it to a simmer.
Add the chopped broccoli, cover, and steam for 3 to 5 minutes until tender-crisp.
Transfer the steamed broccoli to a food processor and pulse a few times until it's chopped into very small pieces.
Combine and Assemble: Add the chopped broccoli to the mashed potato mixture and stir to combine.
Divide the mixture evenly among eight ½-cup ramekins and smooth the tops.
Prepare Topping and Broil: In a small bowl, mix the whole wheat bread crumbs and olive oil together. Spoon this mixture evenly over the gratins.
Place the oven rack about six inches from the top heating element and turn on the broiler.
Place the ramekins on a baking sheet and broil for 30 to 60 seconds, watching carefully, until the topping is golden brown and crisp.
Serve immediately.
Add a Cheesy Boost: For a savory, dairy-free cheesy flavor, mix 1 to 2 tablespoons (30 ml / 1 fl oz) of nutritional yeast into the potato mixture along with the other seasonings.