Heat a very large skillet or a Dutch oven over medium-high heat and coat it with about 3 Tbs. of the olive oil; reserve the rest for the sauce. Add the potatoes and sauté until they start to brown a bit around the edges, about 10 minutes. Raise the heat to high and add the eggplant, peppers, chile, tomatoes, and onion. Season with a little salt. Continue to sauté a few more minutes. Add about 1/4 cup water, cover the pan, and turn down the heat to a vigorous simmer. Braise until all the vegetables are tender when poked with a knife, especially the potatoes and eggplant, 15 to 25 minutes. During braising, be sure to stir and scoop the vegetables so they don't stick to the pan.
Meanwhile, make the sauce. Using a mortar and pestle or a food processor, smash or purée the garlic with 1/2 tsp. salt, the paprika, cumin, and cayenne to make a smooth paste. Add the parsley and cilantro and pound or process just until they're crushed—don't completely purée them. Add the lemon juice and the remaining olive oil.
If any liquid remains in the pan when the vegetables are cooked, remove the cover, turn up the heat, and boil it off. Then turn off the heat, add the sauce to the skillet, and stir to coat all the vegetables. Taste for seasoning and add more salt, lemon, or cayenne to taste. Serve hot.