Trim ends from fresh yuca and peel remainder, removing all waxy brown skin and pinkish layer underneath.
Cut yuca into 3-inch-thick pieces. Boil yuca in salted water to cover by 2 inches until tender and starting to fall apart, 50 minutes to 1 1/4 hours.
Drain and transfer to a cutting board. Carefully halve hot yuca pieces lengthwise and remove thin woody cores. Return yuca to pot with 1 3/4 cups milk, garlic, juice, and salt. Coarsely mash yuca, adding additional milk if desired, and serve immediately.