Trim and clean Brussels sprouts. Bring salted water to a boil in a pot over medium-high heat. Add Brussels sprouts and cook for 15 minutes until tender. Drain well and transfer to a blender. Add milk and blend until smooth. Season with salt, pepper, and nutmeg to taste.
Peel and dice potatoes and carrots. Bring vegetable stock to a boil in a pot over medium-high heat. Add potatoes and carrots and cook for 15 minutes until tender. Drain.
Season chicken fillets with salt and pepper. Heat oil in a large skillet over medium-high heat. When hot, add chicken fillets and fry for 10 minutes, turning every 2-3 minutes, until cooked through to an internal temperature of 74°C (165°F).
Spoon Brussels sprout puree onto serving plates or a platter. Arrange chicken fillets on top. Add potatoes and carrots alongside the chicken.