Heat the oven to 350 degrees. Cut the fennel bulb in half, then cut each half in 3 wedges. Lay them in a single layer in small baking dish, drizzle them with olive oil, measure in ¼ cup water, sprinkle with ½ teaspoon salt, cover with foil and slide into the oven. Bake until the fennel is tender, about 1 hour. Cool. Remove the fennel to a cutting board, pull off any exterior layers that seem fibrous, cut out and discard the pieces of core that hold each wedge together, then chop the remainder into tiny pieces—the tinier the better here.
While the fennel is cooking, scoop the apple into a bowl, sprinkle it with the lime juice and toss to combine. Refrigerate until you’re ready to use it.
Mash the avocado flesh in a large bowl: Cut the avocados in half, running a knife around the pit from top to bottom and back up again. Twist the halves in opposite directions to release the pit from one side. Scoop out the pit, then scoop the flesh from each half. With an old-fashioned potato masher, large fork or back of a large spoon, coarsely mash the avocado.
Add the thyme, chopped green chile, apples (including all the lime juice) and half of the chopped fennel to the avocado, and stir to combine. (Refrigerate the remaining fennel to add to a salad or pasta dish.) Taste and season with salt, usually about 1 teaspoon. Cover with plastic wrap directly on the surface of the guacamole and refrigerate until serving time. When you’re ready for the guacamole, scoop it into a serving dish and sprinkle with the cheese.