Meanwhile, cut the top and bottom off of the fennel bulb, chop in half, and remove the core. Thinly slice the fennel. (Mandolin is great here.)
Add the oil to the pan and wait about 30 seconds, then add the fennel. Spread it out and brown it, stirring occasionally, for about 15-18 minutes.
When the fennel is nicely browned, add the garlic and the asparagus, and saute until the asparagus is bright green but still crisp, about 3 minutes. Add balsamic and season with salt and pepper to taste, stirring and cooking for about 30 seconds more.