Heat the butter and oil over medium heat until the butter melts. Add the onions and carrots and cook, stirring occasionally, until the onions are translucent, about 5 minutes. Add the garlic and cook, stirring, for 30 seconds. Add the flour and cook, stirring for 1 minute.
Stir the chicken broth into the pan and cook, stirring occasionally, until the mixture thickens, about 8 minutes. Add the chicken, peas, and pearl onions, and cook just until the mixture is heated through, about 3-4 minutes. Season to taste with salt and pepper.
Divide the mixture among 4 (2-cup) oven-proof bowls or ramekins.
Combine the cornbread baking mix, milk, oil, and eggs in a mixing bowl. Stir until fully combined. Divide the mixture among the four bowls, spreading into an even layer. Bake for 20-25 minutes or until the cornbread is golden brown.