You look through the viewport of the spacedock and you see her - the only ladle-class (technically, Caduceus class) that can look cool. It can only look cool from one angle - the top - but still. It's an achievement.
From the side, she looks like any other ladle - a big round bowl, with warp engines and nacelles at the wide blunt end of a 'handle'. There's a hint of gold-brown - bronze-coloured - paint on the middle of the handle, and maybe a gold circle on the top of the bowl.
Viewed from the top, though, you can see how she gets her name. Kol-Ut-Shan is the name of the Vulcan principle of Infinite Diversity in Infinite Combinations - IDIC - and has a symbol of the same name - a triangle with a small circle at its tip, angled into a larger circle. From the top, U.S.S. Kol-Ut-Shan's markings look exactly like, well, the Kol-Ut-Shan. The triangle is bronze, the circle gold, and the small inner circle is green - for Vulcans, the colour of life and passion.
Despite the ship's derogatory nickname of Plomeek Ladle, though, she's not a Vulcan ship.
The name and paint job are courtesy of her first captain, a Vulcan who's since retired, and you've heard that now the captain is an Orion. What carries on, though, is that it's a very fitting moniker as well as a cool paintjob. She has the most diverse crew in Starfleet. Almost every ship has a majority species, a crew that's over 75% Human or Andorian or Vulcan or what have you. It just makes sense - with different needs and different cultures, building a ship that works for everyone to work together in is difficult. That doesn't mean that it's impossible. Kol-Ut-Shan proves it. She's the only ship where no species makes up more than 25% of the crew, or more than 30% of the commissioned officers.
She's a medical research and outreach ship, and she stands for everything that the Federation should.