Cook shallots with oil in a 10-in. frying pan over medium heat, stirring occasionally, until deep golden brown, 10 to 20 minutes; reduce heat if they start to brown unevenly. Lift shallots from oil with a slotted spoon and drain on paper towels. Set oil aside.
Meanwhile, slice zucchini into 1/4-in. rounds and quarter pattypans. Put squash in a colander, sprinkle with salt, and gently toss to coat. Set aside 30 minutes to drain.
Stir lime zest and 3 tbsp. juice, the brown sugar, fish sauce, garlic, and ginger together in a medium bowl until sugar dissolves.
Blot zucchini dry between layers of paper towels. Add zucchini and cilantro to bowl with lime juice mixture and toss to coat; taste and add remaining 1 tbsp. lime juice if you like. Transfer to a shallow serving dish and top with shallots. Drizzle reserved oil over salad.