Preheat the oven to 450 degrees F. Line a sheet pan with parchment paper.
Cut the peppers in half lengthwise, then remove the seeds and white pith. Place the peppers cut-side down on the prepared sheet pan.
Roast the peppers for 30-40 minutes, until the skin is blistered and charred.
Carefully transfer the peppers in a large bowl and cover with a clean dish towel. Let sit for 15-20 minutes to loosen the skins. Once the peppers are cool enough to handle, remove and discard the skins.
Place the roasted peppers and any juices from the bowl in a blender or food processor, along with the lemon juice, salt, pepper, and smoked paprika. Blend or process until smooth.
Season the chicken thighs all over with salt and pepper.
Heat the olive oil in a large skillet set over medium heat, then add the chicken thighs and cook for 5-7 minutes per side until cooked through (165 degrees F on an instant read thermometer) and slightly browned, adjusting the heat as needed. Transfer to a plate and cover to keep warm.
Reduce the heat to medium-low. Add the shallots and cook for 2-3 minutes, stirring and scraping up any brown bits on the bottom of the pan. If it looks like the bits might start to burn at any point, add a splash of water and scrape them up using a spatula.
Add the garlic and stir for about 60 seconds until fragrant.
Add the red pepper mixture to the pan along with the cream and basil. Bring to a simmer and cook for 2 minutes, stirring, then add the chicken back to the pan, spooning the sauce over the chicken. Taste and season with additional salt and pepper if desired.
Garnish with additional sliced basil before serving.