Put a medium saucepan over medium-high heat and add 2 tablespoons olive oil. Once the oil is hot, remove the pan from the heat and add the wheat berries, then return to the heat. Add a pinch of salt and allow the berries to toast. Add 3 1/2 cups water and cook until the wheat berries start to break apart and the water is mostly absorbed, about 45 minutes. Taste; they should be chewy but firm. Set aside.
Meanwhile, bring a large pot of water to a boil over medium heat. Add salt until the water tastes like mild seawater. How will you know? Taste a drop of water.
Prepare an ice bath by filling a large bowl with cold water. Add some ice cubes and line the bowl with a strainer. Stir the green beans into the ice bath within the strainer to awaken any beans that are limp. Strain and add them to the boiling water; cook 2 to 3 minutes. Use a slotted spoon or spider to remove the beans from the water, immediately transferring them back to the ice bath. Allow them to sit in the ice bath for a few minutes, swirling them gently.
Transfer the green beans to a kitchen towel and pat dry. Transfer them to a serving bowl and refrigerate.
In a medium bowl, whisk the remaining 2/3 cup olive oil, the sherry vinegar and balsamic vinegar. Season with salt and pepper and set aside. When ready to serve, add the wheat berries to the beans and drizzle with the vinaigrette to taste. Toss to combine and season with salt and pepper, if needed. Serve immediately.