Rinse the barley under cold water and drain. Bring 7 cups of water to a boil in a 4-quart pot over high heat. Add 3/4 tsp. salt. Add the barley, reduce the heat to a simmer, and cook uncovered, stirring occasionally and adding more boiling water as necessary to keep the barley covered, until tender, about 30 to 45 minutes. Drain and rinse the barley with cold water to stop the cooking.
Transfer the barley to a foil-lined rimmed baking sheet, drizzle with 1 Tbs. of the olive oil, and toss lightly to coat. Spread the barley on the baking sheet and cool completely at room temperature or in the refrigerator.
Put the lemon juice in a small bowl and gradually whisk in the remaining 1/2 cup of olive oil. Whisk in the lemon zest. Taste and season with salt, pepper, and additional lemon juice or olive oil as needed.
Put the cooked and cooled barley in a large serving bowl and toss to break up any clumps. Add the peas, sugar snap peas, asparagus, dill, mint, and 1/2 cup vinaigrette and toss. Taste and season as needed with more vinaigrette, salt, pepper, and serve.
The salad can be refrigerated for up to 1 day. If making ahead, let sit at room temperature so it’s not refrigerator-cold and season with more vinaigrette, salt, and pepper before serving.