Heat ½ tablespoon olive oil in a large saucepan over medium-high heat, swirling to coat pan. Add chopped onion and salt and sauté, stirring occasionally, until golden brown, about 20 minutes. If onions start to cook too quickly, add about ¼ cup water to slow the cooking and allow carmalization. Remove onions from pan to a large bowl, and set aside.
Heat ½ tablespoon olive oil in the same pan and add in the mushrooms. Sauté mushrooms until they release all of their water and start to brown, about 10 minutes. Remove mushrooms and either place in the large bowl with the cooked onions, or place the cooked mushrooms in a food processor and pulse until finely chopped, adding finely chopped mushrooms to the bowl after chopping. Stir water chestnuts and chestnuts into the bowl with the veggies.
In the same pan, melt butter, adding ¼ cup olive oil, and once butter is melted, stir in broth. Heat broth to a low a low boil, and remove from heat. Stir dressing into broth, and mix moist dressing into the bowl with the veggies. Taste dressing for moisture and seasoning. If the dressing seems a bit dry (remember, the dressing will dry out a bit in the oven), add in more broth, mixing to combine. Spoon dressing into large oven-proof casserole dish, and bake for 30-45 minutes, until heated through.
Serve hot, garnishing with chopped chives, green onions, or fresh thyme leaves if desired.
Dressing can be made in advance and refrigerated until ready to use. If you’re putting cold dressing in the oven, you may need additional baking time to heat through.