Bring a large pot of water to a boil over medium-high heat. Season the water generously with salt. Add the pasta and cook until al dente according to the package directions. Drain and set aside. Keep warm.

Meanwhile, heat 2 tablespoons of the oil in a large skillet over medium-high heat. Sprinkle the chicken breasts generously with salt and black pepper on both sides. Carefully add to the skillet and cook, undisturbed, until golden brown on the bottom, about 5 minutes. Flip the chicken with tongs and continue to cook until golden brown on the other side, about 5 minutes more. Remove to a plate.

Add the remaining 1 tablespoon oil and the shallot to the same skillet. Cook, stirring occasionally, until just tender, 1 to 2 minutes. Add the garlic and cook, stirring continuously, until fragrant, about 1 minute more. Add the chicken broth and scrape up any browned bits from the bottom. Add the sundried tomatoes, chicken bouillon cube, heavy cream, oregano, red pepper flakes and ¼ teaspoon salt. Stir until the bouillon cube is dissolved.

Bring the sauce to a boil, then reduce to a simmer over medium-low heat. Nestle the chicken breasts in the sauce. Cook until the sauce has thickened slightly and the chicken is cooked through (registering internally at 165 degrees F), 5 to 10 minutes. Stir in the Parmesan and basil. Divide among 4 dinner plates and serve with the spaghetti.