Make the lemon-garlic sauce: combine lemon zest, lemon juice, minced garlic and toasted walnuts in a blender and blend on high until smooth, about 30 seconds.
Add the grated Parmesan and 3 tablespoons of olive oil to the blender and blend again until well combined; transfer the sauce to a small bowl and set aside.
Cook the spaghetti in a large pot of boiling salted water according to package instructions until al dente; reserve 1 cup of pasta cooking water, then drain the pasta.
Heat 1 tablespoon olive oil in a skillet over medium-high heat, add the capers and cook until crisp, about 3 minutes; transfer to a paper-towel-lined plate to drain.
In the same skillet, add the remaining olive oil and reduce heat to medium; add the cubed sourdough and cook, stirring occasionally, until golden and crisp, 2–3 minutes; transfer to the plate with the capers.
Pat the salmon pieces dry and season with salt and pepper. Melt the butter in the skillet over medium-high heat and add the salmon; cook until just fork-tender and opaque, about 3–5 minutes, then remove from the skillet and set aside.
Add the reserved sauce, cooked spaghetti and up to 1 cup reserved pasta water to the skillet and cook briefly over medium heat, tossing with tongs to coat the pasta evenly; add more pasta water as needed to reach desired consistency.
Gently fold in the cooked salmon and watercress until combined and the watercress begins to wilt.
Serve the pasta topped with the crispy capers and toasted bread cubes and additional watercress if desired.