I am no expert on carrots per se, but I wrote a book on Dutch tulip cultivation in this period, so I can add a bit of information from the botanical side of things.

"Breeding" specific varieties of plant was not practical in the 16th or 17th centuries because the science that underpinned it was not understood. Gardeners who wished to produce crimson flowers, for example, might attempt to do so by pouring dark red wine over their tulip beds; and there were attempts to produce flowers with red-and-white petals by cutting the bulb of a white-petalled flower in half and strapping it to half the bulb of a red-petalled flower. So cultivation was a slow and somewhat haphazard business. New cultivars seem to have emerged by accident rather than design.

With that said, there certainly was awareness that the seeds or bulbs of a specific variety of plant would normally produce offspring that were identical to the parent, and so what the horticulturalists of the period certainly were capable of was pouncing on a natural mutation – a yellow carrot which was far more orange in colour than its fellows – isolating it, and carefully collecting and separately planting its seeds. Any carrot that mutated in such a way as to contain higher amounts of betacarotene than the standard yellow, purple and white carrots of the early modern period would meet this requirement. From this perspective, it's not at all implausible that the Dutch might have fairly rapidly succeeded in cultivating more orange strains of carrots, and there are, as you note, numerous cookbooks and other secondary sources that do make this claim.

I've not managed to find a work by a food historian or botanist that traces the claim back to contemporary sources, though, so it might equally be a case of a legend attaching itself to the carrot at a much later date.

Source

Mike Dash, Tulipomania (1999)