Heat the fat over medium-high heat in a large, deep, pan. I use a nonstick one, but enamel cast iron would work great too. Once the fat is melted and hot (but not smoking or burning) add the onion, celery and bell pepper. Cook, stirring frequently, until the veggies are soft, about 10 minutes.
Add the garlic and stir to incorporate until fragrant, about two minutes. Then add the eggplant and mushrooms and continue to cook over medium-high heat until they have released their water and the eggplant browns slightly. Stir by lightly folding, trying not to turn the eggplant into complete mush as it cooks.
Once the eggplant and mushrooms are cooked sufficiently add the tomatoes, balsamic, capers, olives, thyme, red pepper flakes, salt and pepper. Bring the mixture to a light bubble over medium heat and let simmer uncovered for 10-15 minutes. The liquid should mostly be cooked off. Taste and season with additional salt and pepper to your preference.
Enjoy warm or pop it in the fridge and enjoy tomorrow cold or heated up.