Put the potatoes in a medium saucepan; cover with cold water and season with salt. Bring to a simmer over medium-high heat and cook until fork-tender, about 5 minutes. Drain and transfer to a medium bowl; drizzle with the wine and let cool. Reserve the saucepan.
Meanwhile, bring a separate saucepan of salted water to a boil. Fill a bowl with salted ice water. Add the haricots verts to the boiling water; cook until crisp-tender and bright green, 2 to 4 minutes. Drain and immediately plunge into the ice water to cool; drain and pat dry.
Place the eggs in the reserved saucepan and cover with cold water by about 1 inch. Bring to a simmer over medium-high heat, then cover, remove from the heat and let stand, 10 to 12 minutes. Drain, then run under cold water to cool. Peel under cold running water.
Make the dressing: Whisk the vinegar, shallot, mustard, thyme, 1/2 teaspoon salt, and pepper to taste in a bowl. Whisk in the olive oil in a slow, steady stream until emulsified.
Toss the tomatoes in a small bowl with salt and pepper to taste.
Add about 1/4 cup dressing to the potatoes and toss.
Quarter the hard-cooked eggs
Divide the lettuce among 4 plates
Arrange the potatoes, haricots verts, radishes, hardcooked eggs and tuna on top.
Pour any juices from the tomatoes into the dressing, then add the tomatoes to the plates.