Put chicken and buttermilk in a large Ziplock bag. Seal and refrigerate 2 to 24 hours, turning bag occasionally. Drain chicken in a colander; discard buttermilk. Don't pat chicken dry. In a clean plastic bag, combine flour, salt, pepper, and thyme. Add several pieces chicken, seal bag, and shake to coat well. Remove chicken to a plate. Repeat with remaining chicken. Heat oil in 1 large or 2 medium sized heavy skillets over medium heat until hot but not smoking. Add chicken, cover and cook, turning occasionally with tongs, 20 minutes, reducing heat slightly if chicken begins to brown. Uncover, increase heat to medium-high and cook, turning often, 10 minutes or until chicken is golden and cook through. Drain on paper towels.
NOTE: Leftover oil may be strained through a strainer lined with a paper towel or a coffee filter and used once more for cooking.
