Cut potatoes into 1-inch cubes and carrots into 1/2-inch pieces, setting them aside. Bring 6 cups of water to a boil in a large pot and add a generous pinch of salt. Once boiling, add the cubed potatoes and cook for 5 minutes until they begin to soften but are still firm. This staggered timing prevents the vegetables from becoming mushy—potatoes need a head start since they take longer to cook than carrots.
Add the diced carrots to the pot with the partially cooked potatoes and boil together for 8-10 minutes until both vegetables are tender but not falling apart. Drain the vegetables in a colander and spread them on a baking sheet or wide dish to cool to room temperature—this stops the cooking process and prevents them from becoming mushy. While the vegetables cool, you can prepare the fried chicken in the next step.
While the vegetables cool, cut the chicken into 1/2-inch bite-sized pieces. Heat the cooking oil in a large skillet over medium-high heat until shimmering. Working in batches if needed to avoid overcrowding, fry the chicken pieces for 5-7 minutes until golden brown and cooked through. I always drain the fried chicken on paper towels to remove excess oil, which keeps the salad from becoming too greasy. Set aside to cool slightly.
In a medium bowl, whisk together the mayonnaise, pickle relish, onion powder, and celery seed until well combined and smooth. I always use Hellmann's mayonnaise because it creates the creamiest, richest dressing that coats the vegetables beautifully. Taste and adjust seasoning with salt and black pepper as needed—remember the chicken and vegetables will have their own salt, so season conservatively.
Once the cooled vegetables are at room temperature, transfer them to a large mixing bowl. Pour the dressing from Step 4 over the vegetables and fold gently until all pieces are coated evenly. Add the cooled fried chicken pieces from Step 3, the pineapple (with its juices for extra flavor), and the diced green onions. Fold everything together gently but thoroughly, being careful not to break up the potatoes or chicken. Chill for at least 30 minutes before serving to let the flavors meld.