Soak beans overnight in a medium bowl with 1 qt. water; or boil in a medium pot with 1 qt. water 2 minutes, then remove from heat and let stand, covered, 4 hours.
Halve fennel lengthwise, then thinly slice lengthwise, discarding core. In a medium saucepan, sauté fennel with 1 tbsp. oil over medium heat until edges brown, 7 to 10 minutes. Pour into a bowl; chill until used.
Add remaining 1 tbsp. oil, the carrot, and onion to pan; saute until starting to brown, about 8 minutes. Add beans and their liquid, cover, and bring to a boil. Reduce heat and simmer until tender, 1 1/2 to 1 3/4 hours, stirring occasionally and adding a little more water if beans start to look dry. Stir in salt, 2 tsp. thyme, and the fennel slices; return to a simmer and cook about 10 more minutes.
Chop enough fennel fronds to make 1 tbsp. and stir into beans with remaining 1/2 tsp. thyme.
*Buy from nativeseeds.org, ranchogordo.com, or anasazibeans.com