In a large high-sided skillet or, if you've got one, a woklike stir-fry pan, warm the olive oil over med-low heat.
Toss in the garlic and cook for 1 minute.
Add in the onion and leek, turning them in the oil, and season with salt and pepper.
Cook, stirring gently, just until the onion and leek soften slightly, about 5 minutes.
Stir in the potatoes and carrots, then pour in the broth.
Toss in zest and lemongrass.
Increase heat and bring the broth to a boil, then lower heat and simmer gently, uncovered, until the vegetables are just short of tender, about 10 minutes.
*You can make the pot-au-feu to this point up to 3 hours in advance; cover and refrigerate; when ready to continue, bring the broth back to a boil, lower the heat so that it simmers, and cook gently until everything is heated through.
While the vegetables are simmering, bring a skillet of water to a boil--you'll use the pan to reheat the eggs right before serving.
With the pot-au-feu at a gentle simmer, drop in the asparagus and shiitakes; cook until tender, about 4 minutes.
Scatter the spinach over the vegetables and cook another 2 minutes, just until slightly wilted.
Taste for salt and pepper; adjust.
Slip the eggs into the skillet of simmering water and give them just a couple of minutes to warm while you spoon out the pot-au-feu.
Ladle the vegetables and broth into shallow soup plates, scatter with chopped fresh herbs, and finish each with a poached or boiled egg; serve immediately.