Slice off the ramp roots and green tops. Set tops aside for later. Finely chop the ramp whites and place them into a medium bowl. Add lemon juice and salt, then whisk in egg yolk to combine.
Very gradually, add the olive oil in a slow, steady stream, whisking all the while.Taste and add more lemon juice or salt if necessary. Set aioli aside (can be prepared ahead of time and refrigerated.)
Blanch dandelion greens in a large pot of boiling salted water until ribs are tender, 4 to 5 minutes; drain in colander. Rinse under cold water to stop cooking and drain well, gently pressing out excess water.
Heat 1 tablespoon butter in a large nonstick skillet over medium heat. Sauté chopped shallot until softened and starting to brown. Toss in dandelion greens and ramp tops; season with salt and pepper.
Cook until any liquid given off by greens evaporates (about 5 minutes); season with salt and pepper. Remove to plate and tent with foil while you cook the trout.
Pat trout fillets dry, then season with salt and pepper. Wipe out skillet, add 1 tablespoon olive oil and heat over medium-high heat. Add trout to skillet, flesh side down. Cook until just opaque in center, about 2 minutes per side.
To serve, divide greens and trout fillets among plates. Top trout with a dollop of ramp aioli and serve with lemon wedges.