Cut each chicken breast into 4 or 5 large chunks; season with salt and pepper. Heat a large nonstick skillet over medium-high heat. Add the olive oil. Add the chicken and cook, turning, until browned on all sides and almost cooked through, 4 to 6 minutes. Remove to a plate.
Add 1 tablespoon butter to the skillet and reduce the heat to medium. Add the carrots and 1/2 cup water. Cover and simmer until the carrots are crisp-tender, 3 to 4 minutes. Uncover and increase the heat to medium high. Cook until the water evaporates, 1 minute.
Reduce the heat to medium and melt the remaining 1 tablespoon butter in the skillet. Add the asparagus, snow peas and 1/4 teaspoon salt. Cover and cook until the vegetables are crisp-tender, about 3 minutes. Return the chicken to the skillet along with the apricot preserves, mustard, scallions, tarragon and lemon zest. Simmer, tossing, until the preserves are melted and the chicken and vegetables are glazed; season with salt and pepper.