In a medium bowl, mix together the spinach, mozzarella, and 1 clove minced garlic. Season with salt and pepper. Set aside.
Place a chicken thigh between 2 sheets of plastic wrap and pound with a meat mallet, rolling pin, or heavy pan until about ¼ inch (6 mm) thick. Season on both sides with salt and pepper and repeat with remaining thighs.
On a clean sheet of plastic wrap, lay 2 slices of prosciutto vertically, overlapping slightly. Lay a chicken thigh on top. Spread a quarter of the spinach filling evenly over the chicken. Starting from the bottom, tightly roll up. Wrap the chicken roll tightly in plastic wrap to secure. Repeat with the remaining ingredients. Chill in the refrigerator for 30–60 minutes, until firm.
Meanwhile, turn the broiler to high.
Cut the bell pepper in half and remove the seeds and membranes. Place cut-side down on a prepared baking sheet.
Broil until the pepper starts to char and turn black, 2–3 minutes. Transfer to a bowl, cover with plastic wrap, and let sit for 5 minutes to soften. Use a paper towel to gently rub off the charred bits and skin, then roughly chop. Set aside.
Turn the oven temperature to 400°F (200°C).
Heat the olive oil in a large skillet over medium-high heat. Add the chicken rolls and cook for 2–3 minutes per side, until the prosciutto is evenly browned, securing with toothpicks if needed. Transfer the chicken rolls to a baking sheet.
Bake the chicken rolls for 15–20 minutes, until the internal temperature reaches 165°F (75°C). Remove from the oven and let rest for 5 minutes.
Bring a large pot of salted water to a boil. Add the fettucine and cook according to the package directions until al dente. Reserve 1 cup (240 ml) of pasta water, then drain and set aside until ready to serve.
Heat the skillet used to cook the chicken over medium-low heat. Add the roasted red pepper and remaining 2 cloves minced garlic and cook until fragrant, 1–2 minutes.
Add the lemon juice and stir to scrape up any browned bits from the bottom of the pan. Add the chicken stock and bring to boil.
Transfer the red pepper mixture to a high-speed blender and blend until smooth.
Pour the sauce back into the pan and bring to a boil over medium heat. Reduce the heat to medium-low and simmer until reduced by about half, about 5 minutes.
Whisk in the cold butter, 1 tablespoon at a time, until the sauce is emulsified. Season with salt and pepper to taste.
Add the cooked fettuccine to the sauce and toss to coat, adding some of the reserved pasta water as needed to loosen the sauce.
Thinly slice the chicken rolls.
Divide the fettuccine between serving plates, top with the chicken, and garnish with parsley and pepper.