In a bowl, mix olive oil, paprika, cumin, garlic, lime juice, orange juice, salt, and pepper. Add chicken and toss to coat. Marinate at least 1-2 hours, preferably overnight for best flavor.
Grill or sear over medium-high heat until charred and cooked through (5–6 minutes per side). Let rest, then slice thinly.
Boil spaghetti in well-salted water until al dente. Reserve ½ – ¾ cup of pasta water, then drain.
In a large saucepan, melt butter over medium heat. Add garlic and sauté for 30 seconds. Stir in flour and cook for 1 minute to form a roux.
Slowly whisk in warm milk, stirring constantly until smooth and thickened (about 5 minutes). Reduce heat and stir in Oaxaca and cotija cheeses until fully melted. Season with salt, pepper, and paprika.
Add cooked pasta and toss to coat. If needed, loosen with reserved pasta water for a creamy, saucy texture.
In a small bowl, mix tomatoes, cilantro, olive oil, lime juice, and salt. Taste and adjust seasoning.
Plate creamy queso pasta in shallow bowls. Top with sliced pollo asado. Spoon pico de gallo over the top and garnish with extra cotija, cilantro, and cracked pepper.