Fill a large saucepan 3/4 full with salted water; bring to a boil. Add beets and cook 10 to 12 minutes, until almost tender. Add carrots and parsnips and cook 5 more minutes. Remove vegetables and let cool.
Preheat oven to 375°. Heat 2 tablespoons olive oil and garlic in a large sauté pan over medium-high heat; cook pepper and mushrooms until wilted, 6 to 7 minutes. Season with salt and pepper. Remove from heat and set aside.
Place 1 chicken breast between plastic wrap and carefully pound until breast is 1/4-inch thick. Repeat with remaining chicken. Remove plastic wrap and spread 1/4 of pesto on chicken breast. Add 1/4 of mushroom-pepper mixture on top. Carefully roll up chicken (jelly roll fashion); secure with toothpicks. Repeat with each breast. In a large sauté pan, heat 2 tablespoons olive oil and sauté rolled breasts over medium-high heat until golden, turning to brown all sides, about 15 minutes. Transfer to a baking sheet and place in oven 10 to 13 minutes.
Over medium-high heat, sauté 2 tablespoons olive oil and vegetables just until hot, 3 to 4 minutes. Season with salt and pepper.
Remove chicken from oven and allow to cool 10 minutes. Remove toothpicks and slice crosswise into 11/2-inch slices. Plate 3 to 4 slices per person with baby vegetables.