In a measuring cup, whisk together the chicken broth, chipotle, adobo sauce, tomato paste, and brown sugar.
Sprinkle the shrimp with a scant 1/4 tsp. salt and a few generous grinds of black pepper. Put a 12-inch skillet (not nonstick) over medium-high heat for 1-1/2 minutes. Add 2 Tbs. of the oil and once it’s shimmering hot, add the shrimp in a single layer. Cook undisturbed until the shrimp browns nicely, about 2 minutes. Flip and brown the second side, about 1-1/2 minutes. Transfer to a large plate. The shrimp should still be a little undercooked.
Add the remaining 2 Tbs. oil and the chorizo to the pan and cook, tossing, until it starts to brown, about 1 minute. Add the zucchini, onion, and pepper, sprinkle with 1/2 tsp. salt, and cook, tossing often, until the zucchini browns in places and is just tender, about 4 minutes.
Add the broth mixture to the skillet and bring to a boil. Reduce the heat to medium low. Stir in the shrimp, about half of the cilantro, and the lime juice. Cook, stirring often, until the zucchini is tender and the shrimp are opaque throughout (cut one in half to check), 2 to 3 minutes. Season to taste with salt, pepper, and more lime juice. Serve immediately, sprinkled with the remaining cilantro.