In a large pot, add the diced potatoes and carrots, and cover with cold water. Bring to a boil over medium-high heat. Add a pinch of chicken bouillon or salt to the water for seasoning.
Cook for 15-20 minutes, or until the potatoes and carrots are fork-tender but not mushy. The carrots should be slightly tender but still hold their shape.
Drain the potatoes and carrots in a colander and let them cool for a few minutes. You can spread them out on a baking sheet to speed up the cooling process.
Once the potatoes and carrots have cooled slightly, add them to the bowl. the mayonnaise, yellow mustard, white vinegar, sweet relish, minced onion, garlic powder, and onion powder. Gently stir to coat the vegetables evenly with the dressing.
For creamier potato salad, mash a few potato pieces while mixing.
Add the chopped hard-boiled eggs to the mixture and carefully fold them into the salad, making sure not to break up the eggs too much. Taste the salad and adjust the seasoning. Add more salt, pepper, or mustard as desired.
Cover the potato salad and refrigerate for at least 1 hour to allow the flavors to meld together. Before serving, sprinkle with paprika for color and a touch of smoky flavor, if desired.