Heat a large deep skillet or Dutch oven over medium-high heat and add 1 tablespoon of the olive oil.
Toss the chicken pieces with 1 teaspoon Italian seasoning, 1 teaspoon paprika, and 1/2 teaspoon of the kosher salt. Add the chicken to the hot pan and sauté, stirring occasionally, until golden and cooked through, 4 to 6 minutes. Remove the chicken to a plate.
Reduce the heat to low and add the remaining 1 tablespoon olive oil. Add the diced shallot and cook, stirring, until softened, about 3 minutes.
Add the minced garlic, black pepper, the remaining 1 teaspoon Italian seasoning, remaining 1 teaspoon paprika, and the remaining 3/4 teaspoon kosher salt. Cook and stir about 30 seconds until fragrant.
Stir in the orzo and the cherry tomatoes and cook, stirring constantly, for 1 minute to coat and lightly toast the orzo.
Pour in 3 to 4 cups of low-sodium chicken broth and increase heat to bring to a gentle boil. Stir frequently and scrape the bottom of the pan to prevent sticking, then cover and reduce heat to medium. Cook, lifting the lid occasionally to stir and scrape the bottom, until most liquid is absorbed and orzo is al dente, 12 to 14 minutes, adding splashes of broth if it becomes too dry.
Stir in the grated Parmesan, baby spinach, red pepper flakes, half of the sliced basil, and the reserved chicken. Cook, stirring, until the spinach just wilts, about 1 minute. Taste and adjust seasoning with salt, pepper, or red pepper flakes as desired.
Serve hot, sprinkled with the remaining basil and additional Parmesan if desired.