Make the chutney: Combine figs, butter, port, water, lemon zest, bay leaves, cinnamon, sugar, salt, and pepper in a saucepan. Bring to a boil, then reduce heat and simmer, partially covered, until liquid is the consistency of a loose jam, about 40 minutes. Discard bay leaves and cinnamon stick, or reserve for garnish. (Chutney can be refrigerated in an airtight container for up to a week; bring to room temperature, or reheat in a pan over low heat, adding water as needed.)
Roast the pork: Preheat oven to 450 degrees. Rub pork with garlic and oil, and sprinkle with salt and pepper. Arrange onion on a rimmed baking sheet, and place pork on top. Scatter carrots and garlic around meat. Drizzle lightly with oil.
Roast, stirring vegetables halfway through, until a meat thermometer inserted into thickest part of pork registers 145 degrees for medium, 20 to 25 minutes. Let rest for 10 minutes. Slice pork, and serve warm or at room temperature with roasted vegetables and chutney.
Unsliced pork and vegetables can be refrigerated in an airtight container overnight; slice, and serve chilled.