Trim the fennel bulbs of their stalks and bottom root. Halve the fennel lengthwise and slice each half into 1/4-inch thick slices. Peel the potato, halve, and thinly slice. Place the fennel, potatoes, garlic, chicken broth, and milk in a large, heavy pot over medium-high heat. If needed, add enough water to just cover the fennel. Bring the mixture to a simmer and continue cooking until the fennel and potatoes are soft, about 30 minutes.
Drain the fennel mixture. Place the fennel mixture in a food processor and puree. If necessary, place half the mixture in the food processor, pulse to puree, and continue adding the fennel mixture until all the mixture fits in the processor. Continue processing until pureed. Add the mascarpone cheese, salt, and pepper. Mix to combine. Add the mint and process until just combined. Add more salt and pepper, to taste.
Transfer the mixture to a serving dish. If the mixture is a little thin stir a few times and let rest for 5 minutes, then serve.