Preheat the oven to 450°F convection. In a bowl, combine 1 lb ground chicken, 1 tablespoon beef bouillon paste, 2 teaspoons Worcestershire sauce, 1 teaspoon Dijon mustard, 1 teaspoon onion powder, 1 teaspoon garlic powder, 1/2 teaspoon dried thyme, 1/4 teaspoon kosher salt and freshly cracked black pepper. Mix just until combined.
Lay a large sheet of parchment on your work surface and generously dust with flour. Using floured hands, press the chicken into a very thin, even rectangle roughly the same size and shape as the slab of Hawaiian rolls. Lightly flour the top surface.
Brush the entire top surface with whisked egg. Completely cover with panko, gently pressing so it adheres. Generously spray with avocado oil until the breadcrumbs are fully hydrated, not just lightly misted.
Place a second large sheet of parchment over the chicken and carefully flip the entire cutlet. Peel away the parchment from the now-exposed side. Brush with egg, fully coat with panko, press gently and generously spray with avocado oil until fully hydrated.
Lightly spray or brush a large sheet pan with avocado oil. Carefully transfer the cutlet off the parchment onto the prepared pan. Bake at 450°F convection for 15 minutes, or until cooked through; check that the thickest part reaches 165°F and allow additional baking time if needed.
Switch to broil and broil for 4–5 minutes, watching closely, until deeply golden and crispy. Broilers vary, so check early.
Spread about 1 cup caramelized onions evenly over the crispy cutlet. Top with 4–5 oz freshly grated Gruyère and 1/4 cup finely grated Parmesan. Broil briefly until the cheese is melted, bubbly and beginning to brown, watching closely.
Slice the Hawaiian rolls horizontally through the middle, keeping the top and bottom slabs intact. Lightly butter or oil the cut sides and toast until golden.
Spread a generous layer of cold prepared French onion dip over the toasted bottom slab. Carefully place the cheesy chicken cutlet on top. Finish with fresh thyme, if using, and freshly cracked black pepper. Add the top slab and slice into 12 sliders.
Combine beef bouillon paste with hot water according to the package directions for your specific bouillon. Pour the hot broth into small bowls or ramekins and serve alongside the sliders. Dunk as you eat to keep the breading crisp until each bite.