Melt the butter in a wide, deep skillet over medium heat. Add the leeks and cook for 8 to 10 minutes, stirring occasionally, until softened but not browned.
Add the garlic and cook for 30 seconds until fragrant.
Pour in the broth and add the lemon slices. Bring to a bare simmer, with only occasional small bubbles breaking the surface.
Season the swordfish steaks with salt and pepper, then lower them gently into the simmering liquid, making sure they're mostly submerged.
Poach for 12 to 15 minutes, keeping the liquid at a bare simmer throughout, until the thickest part of the fish reaches 140°F on an instant-read thermometer.
Carefully remove the swordfish with a slotted spatula and let rest for 2 minutes.
Divide the leeks and a spoonful of the poaching liquid among shallow bowls.
Top each bowl with a swordfish steak and garnish with fresh dill or tarragon.
Serve immediately while everything is still warm.