Preheat the oven to 425°F (220°C) and line a sheet pan with parchment paper.
Toss the carrots with 2 tablespoons olive oil, 1 teaspoon thyme, salt, and pepper. Spread in a single layer on one side of the pan.
Rub the turkey breast with the remaining olive oil and thyme, season generously, and place on the other side of the pan.
Roast 20 minutes, then check and stir the carrots for even browning.
Continue roasting until the turkey reads 160°F (71°C) internally, another 10 to 15 minutes.
While roasting, make the salsa verde: combine parsley, capers, garlic, red wine vinegar, lemon zest, salt, and pepper, then whisk in the olive oil.
Remove the turkey and carrots from the oven. Tent the turkey with foil and rest 10 minutes.
Slice the turkey, arrange with the carrots, and spoon salsa verde over both before serving.