Pour 2 teaspoons olive oil into an 8- to 10-inch nonstick frying pan over medium heat; when hot, add onion and stir occasionally until slightly limp, about 8 minutes (onion should be a little crunchy still). Remove from heat and let cool.
Meanwhile, in a 3- to 4-quart pan over high heat, bring about 2 quarts water to a boil. Add green beans and cook until tender-crisp to bite, 4 to 5 minutes. With a strainer or slotted spoon, transfer to a colander; rinse under cold running water until cool. Drain and pour into a large bowl. Add carrots to boiling water and cook until tender-crisp, 5 to 6 minutes. Transfer to colander, rinse until cool, drain, and add to bowl. Add peas to boiling water and cook until barely tender to bite, about 3 minutes. Pour into colander and rinse until cool; drain and add to bowl.
In a small bowl, mix lemon juice, yogurt, and mustard until blended. Whisk in remaining 1/4 cup olive oil and salt and pepper to taste.
Add mâche to vegetables in bowl. Drizzle lemon vinaigrette over the top and mix gently to coat. Divide mixture among salad plates and garnish equally with sautéed onion.