Removing eggshell fragments when separating egg yolks and whites is frustrating. Luckily, there are many tools you can use to separate yolks and whites.
A julep-style strainer works well, with holes designed to separate ice chips from cocktail liquid and a convenient handle. A slotted spoon can also do the trick.
Even a funnel can be helpful in a pinch. The egg whites will flow through the opening, leaving the yolk behind in the cone.
A disposable plastic water bottle can also be handy. Crack the egg into a bowl, squeeze the empty bottle, and place its opening directly over the yolk.
When you release your grip, air will be drawn into the bottle, and the yolk will be sucked in as well. Many gadgets employ this same suction method for egg separation.