Heat the oven to 375°F (190°C).
Place the beets in a small lidded pot or a baking dish deeper than the height of the beets. Pour in the broth to come about two-thirds up the sides of the beets. Add the garlic, half the dill, the vinegar, and olive oil and salt to taste. Put the lid on the pot or cover the dish tightly with foil and cook for about 1 hour, depending on the size of the beets, until they are tender all the way through when pierced with a skewer.
When the beets are cool enough to handle, cut them in half and brush generously with olive oil. Meanwhile, with a sharp knife, slice the plums as thinly as you can, cutting around the pit. Tear the remaining dill into pieces, discarding any tough stems, and set aside.
Heat a large cast-iron griddle over high heat and warm the plancha.
Brush the hot plancha or griddle with olive oil. When the oil shimmers, add the beets cut side down. Cook until crisped on the bottom, about 2 minutes. Transfer to a wide platter.
Arrange the plums around the beets on the platter. Add dollops of ricotta, the sliced chiles, the breadcrumbs and the remaining dill.