Looks like it can be done if you get the min and growing and mature before the freeze. I don’t grow carrots because they’re so inexpensive where I live. Wisconsin grows onions, carrots and potatoes in the entire middle of our state where the glaciers ground the soil into a lovely sandy loam. ;)
The TOPS won’t survive the winter, so carrots won’t ‘perk up’ again come spring, but storing them and digging them as needed through the winter months can be done.
https://www.gardeningknowhow.com/edible/vegetables/carrot/overwintering-carrots.htm
Diana in the Driftless, good morning and thank you for this great gardening thread!
https://www.youtube.com/watch?v=fq4JxN8nJcs
or
http://www.ecosnippets.com/gardening/4-day-carrot-growing-technique/
For his plastic mulch method you only need to water one time, when you plant and before you put the plastic down, and gets the carrots sprouted within 4 days, so if water is an issue it might be worth considering.
**Old gardeners/farmers method, boards on the rows!**
When I have grown carrots I tried to get them to germinate as quickly as possible by covering the watered seeded row with some old boards. It conserves moisture and protects from drying sunlight. Water over and around the board daily and in about 4-5 days you should be able to move the boards off and see the germinating carrots. Move some mulch... I used e-z straw, few seeds...around the row and keep watering. Diana might be able to suggest some fast growing carrots and appropriate fertilizer. (Kuroda? Napoli?) Be aware that if you have soil temperatures over 80 degrees Fahrenheit the seeds might not germinate!
(Since I live on a clay soil hilltop with limited space and loam is limited I had to build a box and fill it with soil to grow anything but short type carrots, Chantaray, royal Chantaray. Did for several years and decided to buy my carrots from the store and focus on other vegtables.)