Place rack in upper third of oven and preheat to 450 degrees F. Spread carrots on a baking sheet, drizzle with olive oil, salt, and pepper, then toss to evenly coat. Roast carrots for 15-20 minutes, until fork tender and lightly blistered. Remove from oven and set aside.
Meanwhile, combine Greek yogurt, Dijon mustard, and cayenne in a small bowl. Preheat a Panini press or large skillet to medium high.
On a work surface, lay out bread slices, butter one side of each, then flip over. Spread the second sides thoroughly with the Greek yogurt-Dijon mixture (I find a spoon works best.)
Top 2 of the bread slices with 2 slices of Havarti each. Place carrots in a row on top of each, so that the carrots cover the slices completely. Top carrots with pickles, then flip the final slices of bread over the sandwich tops, butter side up.
Place the sandwiches on the preheated Panini press or skillet and toast until the bread is golden and the cheese is hot and melted (about 3-5 minutes total for a Panini press or 3 minutes per side in a skillet.) Let cool for 2 minutes, slice, and serve immediately.