Heat olive oil, onions, and garlic in a saucepan over medium-high heat. Saute, stirring occasionally until translucent, 6 to 8 minutes. Add tomatoes, salt, oregano, basil, fennel seeds, thyme, and red pepper flakes; stir to combine. Reduce heat; cook over low heat for 15 minutes. Adjust seasoning if necessary; keep warm.
Place chicken on a cutting board. Slice each breast lengthwise into 2 thin cutlets; slice each cutlet into 2 smaller pieces.
Preheat oven to 450 degrees F (230 degrees C). Place a cooling rack on a baking sheet and set aside.
Pour panko and eggs into separate shallow dishes. Combine flour, Parmesan cheese, 1/2 teaspoon salt, 1/2 teaspoon oregano, 1/2 teaspoon basil, and 1/4 teaspoon black pepper in another shallow dish.
Heat olive oil over medium-high heat in a large skillet. Dredge chicken pieces first in flour mixture, then eggs, and finally panko. Pan fry the chicken until lightly browned, no longer pink in the center, and the juices run clear, 2 to 3 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Remove to the cooling rack; sprinkle lightly with salt.
Top each piece of chicken (still on cooling rack) with 2 tablespoons of marinara sauce and 1 tablespoon of mozzarella cheese.
Bake in the preheated oven until cheese is melted, 3 to 5 minutes. Remove from oven; place chicken on buns.
To make chicken Parmesan sliders ahead of time, prepare the chicken and marinara sauce, cool, and refrigerate until dinnertime. Reheat the chicken topped with the marinara sauce and mozzarella cheese in a 400-degree oven for about 10 minutes before serving on buns.