Bring a few inches of water to a boil, salt it and par-boil the broccoli florets to tender-crisp, 3-4 minutes. Cold shock in ice water and drain well, pat dry on kitchen towel.
Heat EVOO, 2 turns of the pan, in a skillet over medium heat. Add chili pepper and garlic, and swirl around for a minute. Add broccoli florets and season with salt and pepper. Toss to combine, turn off heat and cool to handle.
Combine turkey with cooled broccoli and garlic mixture, season with salt and pepper, and form into 6 patties that are thinner at center and thicker at edges for even cooking.