Add the red wine vinegar, garlic, oregano, and a large pinch of salt to a small mixing bowl. Make the dressing by very-slowly drizzling in the olive oil and whisking until emulsified. Season to taste with additional salt.
Bring a medium pot of salted water to a boil. Add the orzo and cook until al dente (about 6 minutes), stirring when it's first added to ensure that it doesn't stick to the bottom of the pot. Drain. In a small mixing bowl, toss the orzo with about 1/4 of the dressing.
Trim away any ragged bits of the chicken breasts, pat them dry with a paper towel, then season generously with salt.
Add the neutral oil to a large cast iron or stainless steel skillet and heat over medium-high heat until shimmering and almost smoking. Swirl the oil to coat the skillet, then arrange the chicken smooth side down in the skillet; cook without moving the chicken, until it is well browned, about 3 minutes. Using tongs, flip the chicken; continue to cook until well browned, about 2 minutes. Reduce heat to medium and continue to cook, turning the chicken once or twice, until an instant-read thermometer inserted into the thickest part registers 160°F or it is no longer pink when cut into with paring knife, 3 to 5 minutes longer. Transfer the chicken to a plate and let rest 5 minutes. Then, slice the chicken breasts and toss with about 1/4 of the dressing.
Meanwhile, toss the spinach, chickpeas, and tomatoes with the remaining dressing.
Divide the orzo, chicken, spinach, chickpeas, tomatoes, olives, and feta between two bowls.
Notes
For a mellower garlic flavor, wait 5-10 minutes before adding the olive oil to the dressing. (The vinegar will help soften the garlic's flavor.)
Recipe by A Modest Feast at http://www.amodestfeast.com/2016/12/06/greek-chicken-chickpea-orzo-bowl/