Preheat the oven to 400ºF / 200ºC degrees and line a baking sheet with foil.
Peel the carrots but leave whole. Place onto a baking sheet and drizzle with olive oil. Season with salt and pepper. Place in the oven to roast for 35–45 minutes until tender.
Make the rub for the pork by mixing together the paprika, brown sugar, garlic powder, onion powder, salt, and pepper. Rub this all over the pork loins.
Place the olive oil in a skillet and heat on medium. Place the pork loins into the skillet and allow to brown on all sides. This will take around 6–7 minutes.
Remove the pork loins to the foil-lined baking sheet.
Take half a cup of the barbeque sauce and brush all over the pork loins then place into the oven for around 15–20 minutes. I like to continue to brush the barbeque sauce over the pork every 5–10 minutes during cooking.
Remove from the oven and allow to rest for five minutes before slicing.
Take a serving platter and lay arugula on the base. Arrange the carrots around the edges of the platter then slice the pork and lay down the center.
Serve with the remaining barbeque sauce.