In a slow cooker, combine carrots, mushroom slices and bay leaves.
In a large bowl, combine flour and thyme. Season chicken with salt and pepper. Add chicken to bowl with flour mixture and toss until evenly coated. In a large skillet, warm 2 tsp. oil over medium-high heat. Add half of chicken and cook, turning once, until golden brown, 3 to 4 minutes per side. Transfer to slow cooker. Repeat with remaining oil and chicken.
Add tomato paste and wine to skillet and cook, stirring and scraping up browned bits, until boiling steadily, about 1 minute. Pour over chicken. Scatter onions on top. Cover and cook on low until meat is tender, 3 to 5 hours.
Remove and discard bay leaves. With a slotted spoon, transfer chicken and vegetables to shallow bowls. Carefully pour sauce from slow cooker into a fat separator. Pour skimmed sauce over chicken; discard fat. Top with baby arugula, if desired.