Heat the oven to 375°F.
Heat 1-1/2 Tbs. of the oil in a 10-inch straight-sided sauté pan over medium-high heat. Add the potatoes and 1/4 tsp. salt and cook, stirring occasionally, until the potatoes begin to brown, about 7 minutes. Transfer the potatoes to a plate.
Add the remaining 1-1/2 Tbs. oil and the fennel to the pan with a pinch of salt and cook, stirring occasionally, until lightly browned, about 5 minutes. Add the onion to the pan and cook, stirring often, until starting to soften and lightly brown, 2 minutes more. Add the broth and bring to a boil. Gently nestle the potatoes into the fennel-onion mixture and add the thyme sprigs. Cover the pan and braise in the oven until the potatoes and fennel are tender, about 20 minutes.
Uncover the pan, add the olives, and continue to braise until most of the liquid has evaporated and the vegetables are meltingly tender, another 10 minutes. Remove from the oven, discard the thyme sprigs, and season to taste with salt and pepper. Garnish with the chopped thyme and serve.