Heat oven to 350 F.
Trim the greens off the top of the fennel bulbs and slice a thin sliver off the root end. Then slice the bulbs into quarters.
Place the fennel in a cast iron pan or skillet with tall sides (something that you an use both on the stove and in the oven), together with 1/3 cup olive oil and 1 cup vegetable broth. Bring it to a boil and simmer for 5 minutes.
Cover the pot loosely with parchment paper and bake in the oven at 350F for 1 hour. Check occasionally to see if there is enough liquid, and you may need to turn some of the fennel pieces.
In the meantime, peel the eggplant, and cut it into 1-inch cubes. Toss the cubes in salt, and place in a colander to drain. After about 30 minutes of draining, pat with paper towels to dry.
During this time, prepare the ingredients for the rest of the stew. Slice the garlic cloves and the chilies. Roughly grate the whole canned tomatoes using a cheese grater with large holes.
Heat a splash of olive oil in a large soup pot then cook the garlic and chilies for 30 seconds or so. Then add the tomatoes. Cook the tomatoes with the garlic and chilies for about 5 minutes. Then add the thyme sprigs, the wine, and the remaining 1 -1/4 cup of vegetable broth. Season with a little salt, bring to a boil, and then simmer around 20 minutes.
Peel the potatoes. Quarter them, then cut the quarters into big wedges around 1.5 inches thick. Steam the potato cubes in a vegetable steamer until soft, around 15-20 minutes.
When the fennel comes out of the oven, it’s time to bake the eggplant. Place the eggplant in a baking dish (I prefer cast iron), and toss with olive oil. Bake at 375F for about 20-30 minutes, or until soft. I usually have the eggplant baking at the same time as I have the potatoes steaming.
Add the roasted fennel and all of its oil juices to the stew. Add the steamed potatoes and the roasted eggplant. Bring the stew back to a boil, and simmer again on very low heat, covered, for 15 minutes.
If you have the time, turn off the heat, and let the stew sit for an hour or so to mellow. Check the seasoning before serving.