Combine all vinaigrette ingredients in a jar and shake well (or whisk together in a bowl) to combine.
If chicken breasts are quite thick, slice them in half lengthwise to thin. (I find that this helps them to cook more evenly and gives you more surface area for the vinaigrette to flavor the chicken.)
Combine chicken with 3 Tablespoons of vinaigrette (reserve the rest of the vinaigrette for the salad), turning the chicken to coat. Refrigerate chicken and let it marinate for at least 20 minutes and up to a day.
While chicken marinates, whenever you're ready to serve, heat up the grill, leaving one part with direct heat and another part over indirect heat. (If you prefer, you can also just cook the chicken indoors using a grill pan or skillet.)
Drain marinade off chicken and season chicken generously with salt and pepper.
Grill chicken over direct heat until grill marks appear. Transfer to indirect heat and continue cooking 4 to 6 minutes, until chicken is cooked through.
Transfer chicken to a cutting board and let rest for 5 minutes (this will help keep it tender and moist). Thinly slice.
In a large bowl, combine greens, cucumbers, strawberries, basil, and feta. Gently toss.
Top salad with chicken and almonds. Drizzle with remaining vinaigrette. Garnish with extra basil leaves. Serve immediately.