A: Welcome to Florida and especially Nassau County. The quick answer is no, we do not have the completely frost free area and mild temperatures required to grow the globe artichokes you mentioned. Ideally, they prefer daytime temperatures between 70 and 75 degrees F. with night time temperatures around 50 – 55 degrees F. We might have temperatures similar in the spring but not long enough for the plant to develop the globe fruit which takes about 80 days.

Globe artichokes, which are part of the Composite family, typically grow in California but I found some information from Texas A&M Extension suggesting home growers in the southern coastal regions of Texas have had some success. The immature globes are surrounded by fleshy flower-like parts called bracts. The base of these bracts are broken off, dipped in a delicious sauce and eaten. Harvesting occurs from March through May.

I am attaching a wonderful article from the University of Illinois regarding some varieties which have been grown in colder Midwest environments. I have also attached a University of Florida publication with other information on the globe and Jerusalem artichoke. That said, the long answer is – it may be possible. You will have to experiment with different varieties to see what might possibly work here. Good luck and keep me posted on your progress – you might just end up being very successful.
http://psas.siuc.edu/faculty/pubs/Walters-71637ArtichokeNews3.pdf
http://edis.ifas.ufl.edu/mv012

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