Add the eggs to a pot and cover them with cold water, making sure the water level is about an inch above the eggs. Set the pot over high heat and bring the water to a rolling boil. Once boiling, remove the pot from the heat and stir in the baking soda to help the eggs cook evenly and peel more easily. Cover the pot and allow the eggs to rest in the hot water for exactly 12 minutes.

While the eggs are resting, prepare an ice bath by filling a large bowl with ice cubes and cold water. After the 12-minute resting period, use a slotted spoon to transfer the eggs immediately into the ice bath. Let them cool completely in the ice water for 5 to 10 minutes to halt the cooking process and prevent a gray-green ring from developing around the yolks.

When the eggs are fully chilled, peel them gently under cool running water—the flowing water helps loosen the shell from the membrane beneath. Dry the peeled eggs with a paper towel, then use a sharp knife to slice each egg in half lengthwise. Gently remove the yolks and place them in a medium mixing bowl. Arrange the empty egg white halves cut-side up on a serving platter so they're ready to be filled.

Add the mayonnaise, relish, yellow mustard, white vinegar, garlic powder, salt, and black pepper to the bowl with the yolks. Use a fork to mash and mix all the ingredients together until the filling is smooth and creamy, being careful not to overmix. Taste the mixture and adjust the seasoning with additional salt if desired to brighten the flavors.

Spoon the yolk filling into a piping bag fitted with a large round or fluted tip for a polished presentation, or simply use a spoon for a more rustic appearance. Pipe or spoon the filling generously into each egg white half, distributing it evenly among all 24 halves.

Lightly sprinkle smoked paprika over the top of each filled egg to add color and a hint of smoky flavor. Cover the deviled eggs and refrigerate until you're ready to serve. They'll keep well in the refrigerator for up to 3 days.