First, go ahead and pre-heat the oven to 450 degrees farenheit.
We'll begin with the peppers - cut the peppers in half and remove the seeds. To an oiled sheet pan, add your peppers, the garlic (skin still on!) and drizzle with a little oil. Rub the oil on the peppers to ensure an even coat.
When the oven is ready, add the sheet pan to the oven and allow the peppers and garlic to roast for about 25-30 minutes. You should start to see some charring on the red pepper skins and they should be wrinkly.
While we wait for the peppers, cook the pasta according to the package instructions.
Separately, in a saute pan, brown the mushrooms with 1 tbsp of butter. They should become nice and brown on the outsides.
Remove the mushrooms and add in the kale. Drizzle with a little oil and cook the kale until it JUST turns bright green and wilts some. On a hot pan, this shouldn't take more than a few minutes.
Now, we'll return to the peppers. Once they're out of the oven, peel the skins off - they should come up very easily. To a blender, add the peeled peppers and garlic, a little salt and pepper and the milk. Blend until smooth. Taste and add more salt or water - for taste and consistency, as needed.
Finally to plating! In a bowl, mix the pasta and vegetables, and add the sauce. Top each with some burrata and freshly cracked black pepper.