Preheat oven to 450 degrees F. Put potatoes in a saucepan with water to cover. Add a large pinch of salt. Cook potatoes until quite tender, about 15 minutes. Drain and put through a ricer or food mill, or mash with a fork or potato masher. Add butter, milk, a few thyme leaves, and salt and pepper to taste. Keep warm in the oven or reheat in a microwave when ready to serve.
While potatoes cook, or immediately after they are done, put an ovenproof skillet large enough to accommodate monkfish over medium-high heat. After a minute, add the oil, garlic, a couple of thyme sprigs and then fillets. Sprinkle with salt and pepper. Cook for about a minute, then without turning, transfer to oven.
Roast until monkfish is cooked through, 8 to 12 minutes depending on its thickness. (It will be tender but still firm when done; a thin-bladed knife will meet consistent light resistance when inserted in its center.) Reheat potatoes if necessary, then serve monkfish browned-side up on top of them, garnished with a little more thyme and pan juices.