Place chicken tenders in a large bowl; sprinkle 1/2 tsp. salt and 1/4 tsp. pepper on top. Add flour and stir to coat chicken.
In a large nonstick skillet, heat half of oil over medium-high heat. Add half of chicken and cook, turning once or twice, until golden on both sides, about 4 minutes. Transfer chicken to a plate and repeat using remaining oil and chicken tenders.
Add onion, curry powder, and remaining 1/4 tsp. each salt and pepper to pan and cook, stirring, until onion softens, about 5 minutes. Add broth and bring to a boil. Reduce heat and simmer until sauce is reduced by half, 3 to 4 minutes. Return chicken to pan and cook, turning, for 2 minutes. Add broccoli and cook, stirring, for 1 to 2 minutes.
Using a slotted spoon, transfer chicken and broccoli to plates. Remove skillet from heat, stir in sour cream and then spoon sauce on top of chicken.