To prepare relish, place cucumber in a colander; sprinkle with salt. Toss well. Drain 1 hour. Place cucumber on several layers of paper towels; cover with additional paper towels. Let stand 5 minutes, pressing down occasionally. Combine cucumber, carrot, vinegar, sugar, and oil in a medium bowl; stir until well blended. Cover and chill 1 hour.
To prepare noodles, cook noodles according to package directions, omitting salt and fat. Drain and set aside.
Combine broth, peanut butter, soy sauce, and chile paste in a small bowl; stir well with a whisk. Heat a large nonstick skillet over medium-high heat; coat pan with cooking spray. Add onions, ginger, and pork to pan; sauté 5 minutes or until pork loses its pink color. Add garlic to pan; sauté 30 seconds. Pour broth mixture into pan; bring to a simmer, and cook 1 minute or until slightly thickened. Pour pork mixture over noodles; toss well. Serve with relish.