1. Put the lentils in a saucepan and add plenty of water. Bring to a boil and simmer for a minute only, then drain. Return the lentils to the pan and pour in just enough water to cover them.
2. Add the bay leaf, onion, and parsley stems, if using. Bring back to a very gentle simmer and cook slowly for about half an hour, until tender but not mushy.
3. Meanwhile, to make the dressing, put the oil, lemon juice, mustard, lemon zest, and sugar in a screw-topped jar, season with salt and pepper, and shake until emulsified.
4. When the lentils are done, drain them well and discard the herbs and onion. While still warm, combine with a good half of the dressing. Leave until cool, then taste and adjust the seasoning; you could add a little more salt, sugar, pepper, or lemon juice if needed.
5. Trim the fennel, removing the tough outer layer (unless the fennel is young and very fresh). Halve the bulb(s) vertically, then slice as thinly as you can, tip to base.
6. Divide about two-thirds of the lentils among wide serving bowls. Scatter over the arugula and fennel and trickle over the rest of the dressing. Scatter the remaining lentils over the top and serve.