In a medium saucepan, combine the potatoes, garlic and enough salted water to cover. Bring to a boil, then simmer until the potatoes are tender, about 15 minutes; drain, reserving 1/4 cup of the cooking water.
Meanwhile, in a medium skillet, heat 2 tablespoons olive oil over medium-high heat. Add the chicken and cook for 4 minutes on each side. Transfer to a plate and shred; season with salt.
In the same skillet, add the onion and thyme and cook, stirring, over medium-high heat for 2 minutes. Add 1 1/2 cups chicken broth and bring to a boil, scraping up any browned bits. Lower the heat to medium, add the peas and carrots and broccoli and cook for 2 minutes. In a bowl, stir the cornstarch into the remaining 1/2 cup broth, then stir into the vegetable mixture. Bring to a boil and cook until thickened. Stir in the chicken and mustard; season with salt and pepper. Keep warm.
In a bowl, mash the potatoes with the remaining 3 tablespoons olive oil and the reserved cooking water. Divide among 4 shallow bowls, make a well in the center and top with the chicken and vegetables.