For the glaze: In a medium bowl, whisk the maple syrup, butter, lemon zest and juice, chile flakes and a pinch of kosher salt.
For the carrots: In a large saute pan on medium to medium-high heat, add the olive oil and when it begins to swirl, add the carrots cut-side down and season with kosher salt and pepper. Cook to sear and caramelize, 5 to 10 minutes, then flip. On the flip, add the glaze and allow it to bubble and reduce, about 2 minutes. Pour everything onto a serving platter. Top with a sprinkle of flakey salt and chopped parsley or carrot greens.