Preheat the oven to 400 degrees F. Place the oven racks in the upper and lower thirds of the oven.
Trim the stems from the broccoli crowns. Slice crowns in half, then slice into stalks, leaving a flat, cut side when possible.
Divide the broccoli stalks between two sheet pans, drizzle with olive oil and sprinkle iwth salt. Using your hands, rub the olive oil into the stalks and florets and rub the oil all around the bottom of the pan. Spread out the broccoli on the sheet pan so there is space between them, with the flat, cut side down. (If they aren't spread out, they will steam, not char.) Place the sheet pans in the upper and lower thirds of the oven.
Check the broccoli after about 12-15 minutes, if it’s nice and charred on the bottom, flip the stalks over, and rotate the pan that was in the bottom of the oven to the top of the oven and vice versa. Continue to cook broccoli until stems are charred and florets are getting crispy, another 5-10 minutes. Total cooking time will be about 23-25 minutes.
While the broccoli cooks, make the lemon yogurt sauce. Put the greek yogurt into a medium mixing bowl. Slice the lemon in half and squeeze all of the juice into the bowl with the yogurt, taking care to strain the seeds out as you do this. Using a microplane grater, grate one clove of garlic into the yogurt mixture.
Finely dice the remaining lemon peel and pulp and add to the yogurt sauce. Add a pinch of salt and stir to incorporate all the ingredients.
Check the broccoli after about 12-15 minutes, if it’s nice and charred on the bottom, flip the stalks over, and rotate the pan that was in the bottom of the oven to the top of the oven and vice versa. Continue to cook broccoli until stems are charred and florets are getting crispy, another 5-10 minutes. Total cooking time will be about 23-25 minutes.
Remove charred broccoli from the oven and allow to cool just enough so you can handle it. Combine the two pans and pour the Calabrian chili paste over the broccoli. Rub it into the stalks and the florets so everything is coated.
Spread the lemon yogurt sauce onto a platter and place the broccoli on top. Scatter crumbled feta over the charred broccoli and serve.