Heat 1.5 cups milk, 3/4 cups sugar, 1/4 cup butter, and lavender buds in a saucepan until sugar dissolves. Simmer for 5 minutes.
Cool mixture, then combine with flour in a stand mixer at a low speed.
Gradually add eggs, starter, and salt.
Increase the mixer speed to medium and knead the dough for about 10 minutes or until the dough pulls away from the edges of the bowl and becomes smooth and elastic.
Cover the bowl with plastic wrap or a damp tea towel and let the dough rise in a warm place for 6-8 hours, or until it has doubled in size. After this initial rise, place the dough in the refrigerator to rest overnight.
The next day, flour a clean work surface and use a rolling pin to roll out the dough to about 1/2 inch thick.
Use a cup or biscuit cutter to cut the dough into circles. To create the classic doughnut shape, poke a hole through the center of each circle and gently stretch it to form a ring.
Place the cut doughnuts on a parchment paper-lined baking sheet. Cover them with a damp towel and allow them to rise for another 2 hours, or until they have puffed up.
In a large skillet or deep fryer, heat your chosen oil (tallow, avocado oil, or coconut oil) to medium heat, about 350°F (175°C).
Carefully place a few doughnuts at a time into the hot oil, making sure each donut has enough room and you aren't overcrowding the pan. Fry each side for about 2 minutes or until they are golden brown and cooked through.
Use a slotted spoon to remove the doughnuts from the oil and place them on a cooling rack lined with paper towels to drain excess oil.