Melt butter in a large, heavy pan over medium heat. Cook and stir onion and garlic in hot butter until tender, about 5 minutes. Whisk flour into the butter mixture until incorporated. Reduce heat to low and cook, stirring constantly, until the liquid is thick, about 3 minutes.
Stir milk, chicken broth, and chicken bouillon cubes into the butter mixture. Increase heat to medium and cook, stirring regularly, until slightly thickened, 5 to 7 minutes. Add mixed vegetables and cook until the vegetables are heated through, 10 to 15 minutes.
Reduce heat to low; add process cheese food, hot pepper sauce, and white pepper. Cook and stir until the cheese is completely melted, about 10 minutes.