Heat the oil in a large skillet over medium-high heat; add chicken and cook until browned, about 3 minutes.
Remove from skillet; salt and pepper to taste. Reduce skillet heat to medium; add onion and cook for 2 minutes. Add remaining veggies and tarragon, tossing well to combine, but don't overcook. Place vegetables in a slow cooker then layer chicken on top; pour broth and wine over all.
Cover and cook on LOW for 5 to 6 hours or until chicken is cooked through. Remove chicken and veggies from slow cooker and keep warm.
Melt the butter in a small saucepan over low heat; add flour and whisk together till well blended (no lumps!); cook for 1 minute, but don't brown. Gradually add liquid from slow cooker; bring to a boil then reduce heat and simmer for 1 minute.
Add half and half; bring to a low simmer and cook until thickened, about 5 minutes. DON’T LET IT BOIL OR THE SAUCE WILL SEPARATE! To serve, pour the sauce over the chicken and vegetables.