Place 6 eggs in a single layer in a saucepan and cover with cold water by at least 1 inch.
Bring to a full boil over high heat then reduce to a gentle boil and cook for exactly 10 minutes for fully set dry yolks ideal for creamy deviled egg filling.
Immediately transfer cooked eggs to a bowl of cold water or ice bath and cool completely for minimum 15 minutes — ice bath stops cooking immediately and makes eggs significantly easier to peel cleanly.
Peel all 6 eggs as carefully as possible keeping egg whites as smooth and intact as possible — any rough spots or tears will be more visible after beet staining — smooth intact surface absorbs color most evenly.
Place peeled eggs in a single layer in a container just large enough to hold them all.
Pour beet liquid over the peeled eggs making sure liquid completely submerges all eggs — add a small amount of water if needed to bring level up enough to cover completely.
Cover container and refrigerate — for light blush pink soak 2–4 hours — for deep vivid magenta soak overnight 8–12 hours — check color periodically and remove when satisfied.
Remove stained eggs from beet liquid and gently pat completely dry with paper towels — any remaining beet liquid on surface will transfer to filling and tint it pink.
Halve each egg lengthwise with a sharp clean knife — wipe blade between each cut for cleanest most precise edges.
Carefully pop each yolk into a mixing bowl keeping pink egg white halves intact — set white halves aside on serving plate.
Add mustard, pickle juice, and mayonnaise to the bowl of egg yolks.
Mash everything together with a fork until completely smooth and creamy with no visible yolk crumbles remaining — run through a fine-mesh strainer for exceptionally smooth restaurant-quality filling.
Season with salt and black pepper to taste — filling should be smooth slightly glossy and just firm enough to hold its shape when piped or spooned.
If filling feels too thick add a very small additional amount of mayonnaise one teaspoon at a time until desired consistency is reached.
Transfer filling to a piping bag with a star or round tip for clean professional swirl — or use a small spoon for rustic home-style presentation — or use a zip-lock bag with small corner snipped off as a quick substitute.
Fill each pink egg white half generously with yolk mixture.
Place filled eggs in refrigerator and chill for minimum 30 minutes before serving — chilling firms the filling and allows flavors to develop and meld.
Just before serving dust lightly with paprika for color contrast and gentle aromatic finish — add paprika only immediately before serving never before storage.
Serve cold directly from refrigerator.