Trim off and discard any browned bits from the ends of the endives and any bruised or browning exterior leaves. You can cut them in half lengthwise if you like.
Put a large, heavy frying pan, sauté pan, or wide pot with a tight-fitting lid over medium-low heat. When the pan is hot, add the butter.
When the butter is melted, lay the endives in a single layer in the pan. If you’ve cut them, lay them cut-side down.
Put the lemon juice in the water and pour the mixture into the pan. Sprinkle the endive with the salt.
If you want to cut the bitterness in the final dish even further than the braising will do alone, sprinkle the endives with the sugar.
Cover the pan. If steam is escaping around the lid, lay a piece of foil over the pan and then put on the lid. Or, cut a piece of parchment paper to fit in the pan directly over the endives and then cover the pan (feel free to do this even with a tight-fitting lid; it helps the endives brown evenly).
Method 1: Reduce the heat to low. Let the endives cook, undisturbed, until they are very tender, about 30 minutes.
Method 2: If your stove is uneven or you need the burner space for the rest of the meal, you can put the pan in a 375°F oven (as long as it's oven-safe) at this point and let them cook in there for 30 to 40 minutes instead of on the stove. Note: This method is more likely to result in more evenly browned endives.
When the endives are very tender, remove the lid (and the foil or parchment paper, if you used it), return the pan to the stove if it's been in the oven, and cook over medium heat until any liquid in the pan evaporates and the endives start to brown, if necessary.
Turn the endives over and cook them until they are browned all over. Note that if you added sugar in Step 6, you'll need to watch them quite carefully at this point, since the added sugar will make them brown more quickly.
Serve the endive hot or warm.