Heat the oil in a large skillet or cast-iron pan over medium-high heat. Add the sliced onion, bell pepper, and mushrooms. Cook, stirring occasionally, for 5-7 minutes until softened and lightly browned. Transfer the vegetables to a plate.
Melt the butter in the same skillet. Add the thinly sliced chicken in a single layer. Cook for 3-4 minutes, turning once, until the chicken is opaque and cooked through.
Return the cooked vegetables to the skillet with the chicken. Add the minced garlic, salt, pepper, and paprika. Stir everything together for about 1 minute until the garlic is fragrant.
Check that the chicken is cooked to a safe internal temperature of 165°F (74°C).
Reduce the heat to low. Arrange the chicken and vegetable mixture into two even piles in the skillet. Drape 2 slices of provolone cheese over each pile.
Cover the skillet with a lid and let the cheese melt, which should take about 1-2 minutes.
While the cheese melts, toast the inside of the hoagie rolls until lightly golden. Spread a thin layer of mayonnaise on the toasted rolls, if you like.
Use a spatula to lift each pile of cheesy chicken and vegetables and place it inside a toasted roll. Serve immediately.