Preheat oven to 425 F. Line 2 large rimmed baking sheet with parchment paper.
Place sliced potatoes on one baking sheet. Drizzle with olive oil (1 tablespoon should do, but 2 tablespoons is ideal for browning.) Sprinkle with 1/2 teaspoon salt and add add several dashes fresh black pepper. Toss with your hands to coat completely. Roast potatoes for 15 minutes.
In the meantime, prepare the fennel and onions. Slice the fennel bulbs from top to base, in 1/2 inch slices. Place on the other baking sheet with the sliced onions and toss with olive oil, 1/2 teaspoon salt and pepper to coat, just like you did with the potatoes.
When the potatoes have roasted for 15 minutes, remove from the oven and use a spatula to flip them. Return to the oven on the upper rack, and place the fennel and onions on the middle rack.
Roast for 10 minutes, then remove the fennel and onions, flip, and roast an additional 5 to 10 minutes. At this point, the potatoes should be tender and the fennel and onions should be caramelized.
Reserve a few pieces of fennel for garnish. If using a food processor, place potatoes and fennel in the workbowl along with warmed milk and broth. Pulse a few times, so that it’s creamy but still chunky. Don’t over-puree or potatoes will turn gummy. Thin further with water, if needed.
You can serve immediately, or transfer to a pot to warm a little more and let the flavors meld.
You can also use a submersion blender by transferring all ingredients to a 4 quart pot and blending so that it’s creamy but still chunky, thinning with water as needed.
Taste for salt and pepper, ladle into bowls, garnish with reserved roasted fennel slices and fennel fronds, and serve!