Preheat oven to 400
Soften Butter ahead of time. Either leave out at room temperature; or, carefully soften in microwave on low setting. Do not melt the butter. Just make it pliable.
Place beets in a deep pan with a little water, a coating of oil, and a pinch of salt. Cover with foil, and roast for about an hour. Check to see if it is done. It should yield to the touch, but, not be too soft. When cooked, cool down, peel, and cut into 1/4 inch or smaller dice.
make an herb butter: Mix the softened butter with herbs, shallot, garlic, chili, Lemon juice, and season with salt and pepper. You can mix by hand or in a food processor to make the herb butter really green.
Pound out the chicken breasts until they are about 1/8 to 1/4 inch thick. Place some of the herb butter in the center of each breast and roll up so the butter is encased in chicken. make sure none is exposed or it will leak out when cooking. Place the stuffed chicken breasts in a freezer for at least an hour. You want them to become firm and maintain their shape when it comes time to bread them.
make the salsa: Mix all salsa ingredients in a bowl. Adjust seasoning as needed.
Bread the Chicken. Once the chicken is solid, Whisk the eggs with a little bit of water. Lightly season some flour with salt and pepper, then set up your breading and frying station: Start with flour, then egg, then bread crumbs. Have oil in a pot or fryer at 350 degrees. And, have a pan lined with paper towels to absorb the extra oil. Dredge the chicken in flour. Coat it evenly, shake off any excess. From there, dip it in the egg. Make sure the egg adheres well. You shouldn't see any chicken or flour exposed. Transfer it to the bread crumbs and make sure the crumbs coat everything. You want to see crumbs all over the chicken, no meat, flour or egg should be visible. This is very important, because if it isn't well encased, some of the butter could leak out during cooking.
Fry the Chicken: Take the breaded chicken and, carefully, put each breast in the fry oil. Try to keep the heat consistent. Cook it until golden brown. This should take about 10 minutes. If too dark, it will taste burnt. Too light, it won't hold together as well, and could be greasy. After drying on paper towels, transfer to a sheet pan and finish in the oven. Cook another 10 to 15 minutes at 400. You can carefully check the internal temperature with a meat thermometer. You want to cook it to 165 degrees.
Allow to rest about 5 minutes before plating. Serve it with the salsa. When you and your guests cut into it, the center will be a nice green and the butter will ooze out, creating a sauce for the chicken.
