Quarter each potato lengthwise, then cut each quarter in thirds. In large pot of boiling salted water, cook potatoes 10 minutes, until crisp-tender; drain and pat dry.
In very large nonstick skillet, over medium-high heat, warm 2 tablespoons of the oil. Add potatoes and 1/4 teaspoon of the salt and cook, until potatoes are golden brown, tossing frequently. Add garlic and cook 2 more minutes. Transfer potatoes to bowl.
Add the remaining tablespoon of oil to skillet. Add chicken and cook 2 minutes, stirring frequently. Add sugar snap peas, tomatoes, the remaining 1/4 teaspoon salt, and the potatoes and cook, until the chicken is cooked through, the sugar snaps are crisp-tender and the tomatoes have begun to collapse, stirring frequently.
In a small bowl, stir together cornstarch and chicken broth. Add to pan along with basil and cook, about 1 minute, until lightly thickened, stirring constantly.