Preheat the oven to 425°. Toss the chicken with 1 Tablespoon olive oil, garlic powder, Italian seasoning, and salt.
Drizzle the remaining 1 Tablespoon of olive in the bottom of a large, oven-safe skillet, like a cast-iron skillet, over medium-high heat.
Once it’s hot and shimmering, pan fry the chicken for 8-10 minutes, flipping halfway through, until golden brown on the outside and the internal temperature reaches just below 165°F. The chicken will continue to cook when it’s baked so it doesn’t need to be fully cooked (i pulled mine off the heat at 145°F). Add a splash of water as needed to the pan if it’s becoming to brown before the chicken is cooked through. Transfer the chicken to a clean cutting board to rest.
In the same pan, add 1 Tablespoon of butter and stir to melt over medium heat, scraping some of the brown bits from the chicken as you go. Add the shallot and cook for 2-3 minutes, until soft and translucent. Stir in the garlic and cook for another 1-2 minutes, until fragrant.
Add the remaining 3 Tablespoons of butter and stir to melt. Sprinkle the flour on top and stir to coat, creating a thick paste (a roux).
With the heat on medium, whisk in the chicken broth, milk, and heavy cream. Turn the heat up to medium-high and continue to whisk until the mixture smooths and starts to bubble and thicken.
Turn the heat off and dice the cooked chicken into bite-sized pieces. Add the chicken to the sauce with the uncooked gnocchi. Stir to coat and sprinkle with mozzarella and parmesan cheese.
Bake in the preheated oven for 12-15 minutes, until the cheese is melted and bubbling around the edges, and starting to turn golden brown in a few places on the top. Check the gnocchi bake 1-2 times as needed in the oven and rotate the pan carefully for even browning.