Trim asparagus spears by gently bending each one; they will snap where the tender stalk becomes woody. (You can save the woody ends for soup stock)
On one parchment-lined baking sheet, toss the asparagus spears with 2 tablespoon of the olive oil, and spread them out across the pan. Sprinkle with pepper.
On the other parchment-lined baking sheet, put the whole peppers, spaced apart
Put both baking sheets into the oven. After 10 minutes, use a fork to check that the asparagus is tender. (if they are very thick stalks, they may need another couple of minutes)
Remove asparagus from the oven and set aside.
Turn the peppers so that they are blistered on all sides. Roast them for a further 10 minutes. The skins should be charred and blistered, and the peppers will be collapsed and tender.
Remove peppers from the oven and immediately put them in a bowl with a lid, or use a plate or foil for a cover. Allow them to steam for 10 minutes
Remove stems from peppers. Cut the peppers in quarters lengthwise and remove skins. Slice the peppers into strips.
Add the sliced peppers to the baking sheet with the olive oil coated asparagus and toss to combine.
Heat 1 tablespoon of olive oil in a skillet over medium-high heat. Fry the halloumi slices 3 minutes per side until golden brown.
Spread arugula on a serving platter. Arrange roasted asparagus and peppers over the arugula.
Scatter with capers, and drizzle with lemon juice.
Top with fried halloumi slices and serve
Notes
Fry the halloumi while the vegetables are in the oven.You can add additional olive oil, but I found that the oil in which the asparagus was roasted was the perfect amount.Halloumi is a brined cheese, making it fairly salty, so I didn't use additional salt. If you are making this recipe ahead, don't use the arugula until serving time. If you can't find halloumi cheese, feta is a great substitute.