Preheat the oven to 400º.

First brown the butter. I do this by placing the butter in a small saucepan over medium heat, and letting it melt. Then keep an eye on it and let it keep cooking. It will foam up and then you will begin to see it start to turn a light brown, first in the center of the pan. Give it a stir every few seconds, and watch it carefully. When the foam subsides it will probably be brown enough, but if it's still too light, let it go another 20 or 30 seconds. It should give off a nice nutty aroma when it is done. It is best to empty the pan into a heat proof dish as the residual heat in the pan can get it too brown.

Put the butter in a large iron skillet or an ovenproof saute pan.

Trim the stem end of each Brussels sprout and cut them in half, laying the cut side down in the pan.

If the shallots are larger than the Brussels sprouts, cut them in half and lay the cut sides down in the pan as well. Try to have as many of both vegetables in a single layer as possible, but some will probably not fit.

Turn the heat to medium. Just when you being to hear the butter sizzle, add the chicken broth and sprinkle the salt evenly over the vegetables.

Cover the pan tightly and allow to cook for 5 minutes at a brisk boil.

Remove the lid and place the pan in the preheated oven.

Check after about 5 minutes and rearrange the vegetables in the pan so that any that weren't touching the bottom before now are.

The chicken broth will cook off - if this is happening too quickly, as in the vegetables are getting too browned, but don't seem soft enough - add the other 1/3 cup of chicken broth.

Cook until the broth has evaporated and the vegetables are well caramelized and tender.

Remove the pan from the oven and stir in the lemon juice.

Put the vegetables in a serving bowl and garnish with the lemon zest - serve immediately