Lightly season the steak with salt and pepper on both sides and set aside.
Warm the oil in a small skillet/frying pan over a medium heat. (Warm a second pan after a couple minutes of cooking if you prefer to cook the steak separately - see below)
Remove the ends from the carrots and radishes. Cut the radishes in to quarters and slice the carrot in quarters lengthways, then each quarter into chunks roughly the same size as the radish pieces.
Add the radishes and carrots to the pan and cook for around 3-5 minutes each side (the three main sides) until lightly brown.
After you turn to the second side, either move the vegetables to the side of the pan and cook the steak in the side of the pan, or cook the steak in a separate pan. If cooking separately, add a little oil or butter to the pan if using a skillet/frying pan, or lightly rub oil on the steak before cooking if using a griddle pan (as I would suggest if doing separately). Cook for a few minutes each side until you achieve your preferred level of doneness then remove from pan to rest.
After the vegetables have been cooking on their last side for a minute or two, create a small space in the middle of the pan and add the egg. Cook for a few minutes until the white is cooked through. If you want a more cooked yolk, baste it with the oil from the pan every minute or two by tipping the pan to the side to gather the oil then spooning over the yolk.
Slice the steak across the grain then place over the vegetables, topped with a few drizzles of the pesto.