Heat vegetable oil in a medium skillet over medium-high heat. Season both sides with salt and pepper.
When oil is just about to smoke, add steak, cook 1 minute on each side.
Reduce heat to medium-low, add butter with fresh thyme and fresh rosemary and cook until internal temperature reaches 140 °C.
Remove the steak and let rest 5 minutes before slicing.
Creamy mushrooms and thyme sauce
Heat the saucepan, over medium heat. Add butter, garlic and finely chopped thyme, cook over low heat for a minute or two to soften garlic.
Add the sliced mushrooms and cook them until they are browned and soft, stirring occasionally, about 5-7 minutes.
Add the fresh cream, season with salt and pepper, simmer and stir for 3 to 5 minutes.
Carrot purée
If you have a steamer, place the carrots, set over boiling water and cook for 15 to 20 minutes until carrots are tender; otherwise place carrots in a medium saucepan, and cover with water. Bring to a boil. Reduce heat, and simmer until carrots are tender, about 20 minutes.
Drain than place in a blender adding salt, pepper, butter and fresh liquid cream until very smooth.