Heat 1/4 cup oil in heavy small skillet over medium heat. Add garlic and sauté until light golden brown, about 6 minutes. Strain oil into glass measuring cup; reserve garlic. Add enough oil to measuring cup to
measure 1/4 cup. Set aside.
Preheat oven to 350 degrees F. Cook linguine in large pot of boiling salted water until tender but still firm to bite, stirring occasionally. Meanwhile, heat reserved oil in heavy large skillet over medium-high heat. Add chicken and sauté 5 minutes. Add poblanos and sauté 2 minutes. Add tomatoes, 1/4 cup basil, prosciutto and 2 tablespoons cooked garlic and sauté until chicken is cooked through, about 1 minute. Remove from heat. Add butter and stir just until melted.
Drain linguine and place in large bowl. Add chicken mixture and 1/2 cup Parmesan; toss to coat. Season with salt and pepper. Sprinkle with remaining 1/4 cup basil and 2 tablespoons cooked garlic. Serve with remaining Parmesan. Serves 4.