I don't think the finished is sprayed. As others have said, it looks like they used a rad roller (very badly).
Pre-primed MDF is normally primed with waterbased paints- they raise the "grain". I you want a super flat finish, you will need to sand them with a fine abrasive (180-240 grit). Unfortunately, the sanding process will cut through the paint at the edges though.
As a a decorator, I prefer to work with un-primed MDF (unless it has been primed with pre-cat spray which doesn't raise the grain). I then apply a coat of Leyland Trade acrylic primer/undercoat. Again, it raises the grain but it is both cheap and very easy to sand flat, easier than the waterbased paints used by timber suppliers. Frankly, I sand most of the paint off until I can see the parallel sanding marks that all mdf has as part of the manufacturing process. See the image below, even though the image is compressed, you should be able to see those parallel lines on the section sanded.
If I were painting the MDF with white oil based eggshell, I would use oil based undercoat to obliterate the dark MDF colour. If it is going to be a darkish colour, I work on the assumption that two coats of oil based eggshell will be fine. The first coat soaks in slightly but when using a decent brand, the second coat provides a uniform sheen.
Decent brushes and paint additives help a lot.
The Purdy Sprig Elite brushes
will help you to apply the paint evenly. They don't hold a lot of paint though, so an additive such as a Owatrol oil
will help the paint flow as you drag the brush over the surface. Additionally, you can add some terebene
to the paint to speed the curing/drying process. You will be using the brush over a number of days- I would recommend buying a BrushMate
to store the (£15) brush. It uses a vapour pad to stop the brush drying out. Ordinarily, I only clean my brushes if I accidently drop them on a dusty floor and they become contaminated.
BTW, the links provided might not be the cheapest- they were the first that I stumbled across.
Best of luck.