In a small bowl, mix together butter, herbs, Dijon mustard, dash of salt, and a dash of pepper. Set aside.
Line the bottom of a 3 quart pot with a tea towel or flour sack. This keeps the egg coddlers from rattling or chipping.
Place jars into the pot and add water. The water should be halfway up the side of the egg coddlers.
Remove the jars. Heat the water on the stovetop over medium heat and bring to a steady simmer (just under boiling).
While the water is heating, grease the sides and bottom of the coddling jars with the butter mixture. Gently crack two eggs into the each jar. Top with a dash of salt and a dash of pepper.
If using the coddling jars, cover with the lid. Use a canning jar lifter to gently place the egg coddlers into the pot of simmering water. Be careful not to allow the water to get into the jar.
Maintaining the simmering water, cook for 9-11 minutes until the egg has reached the desired doneness. If using the mason jars, add 2-3 spoonfuls of simmering water to the top of each jar to poach the top of the egg at about 7 minutes into the cook time.
If water was used or got into the jars, simply drain the liquid off the top before serving.
I like to eat these eggs with gluten-free toast, but you can eat them straight out of the jar. They are delicious!