Trim the ends off the ends of the zucchini and cut on the diagonal into slices 1/2 inch thick.
Place in a colander and sprinkle with salt, tossing to distribute the salt evenly. Spread the zucchini out in the colander and let stand over a bowl, or in the sink, for 40-50 minutes to drain off the excess moisture and bitterness. Rinse and pat dry with a clean kitchen towel. Set aside.
Using a soft brush or a clean kitchen towel, clean the mushrooms of any bits of soil; DO NOT WASH. Trim the stems. Cut the mushrooms into slices 1/4 inch thick. Place in a bowl and set aside.
In a large saute pan or frying pan over medium heat, warm 2 TBS of the olive oil.
Add the zucchini and saute, tossing often, until tender but still firm - about 8-10 minutes. Transfer to a bowl and set aside.
Add another 2 TBS of olive oil to the same pan over medium-high heat. Add the sliced mushrooms and saute, tossing often, until the mushrooms begin to soften - about 6-7 minutes. Transfer to the bowl with the zucchini.
Add the remaining 1 TBS of olive oil to the pan, reduce the heat to medium-low, and add the chopped garlic. Saute gently, stirring, until the garlic just begins to change color - about 30-40 seconds.
Return the mushrooms and zucchini to the pan, reduce the heat to medium-low, and add the basil. Salt and pepper to taste. Raise the heat and toss until the vegetables are hot.
Transfer to a warmed serving dish. Using a zester or a fine-holed shredder, shred the zest (yellow part only) from the lemon directly onto the vegetables and serve immediately.
NOTE: you can substitute grated Parmesan cheese for the lemon zest, if desired.