Season chicken with salt and pepper on both sides and lightly coat with flour, shaking off excess.

Heat olive oil and butter in a large nonstick skillet over medium heat. Add chicken to the skillet and cook until tenders are golden brown, no longer pink in the centers, and juices run clear, 5 to 6 minutes, flipping halfway through. Transfer to a plate and cover loosely with aluminum foil to keep warm.

Add shallot and garlic to the same skillet and cook until softened, about 2 minutes. Add red wine and balsamic and let simmer until reduced by half, 3 to 5 minutes. Stir in figs, thyme, Dijon mustard, honey, and salt; bring to a boil. Reduce heat to low or medium-low and let simmer until thickened, about 3 minutes. Remove from heat and stir in butter.

Spoon sauce over warm chicken tenders, or add chicken back into the skillet to briefly reheat, and serve immediately.