Warm the milk in a small saucepan to about 105 F / 40 C. Whisk in 1 tsp of the sugar and the yeast, then let sit 10 minutes until foamy.
In a stand mixer bowl or large mixing bowl, combine the flour, remaining sugar, salt, lemon zest, and fennel seeds.
Add the yeast mixture, eggs, and softened butter. Mix on low, then knead 8 to 10 minutes by hand or with the dough hook until smooth and elastic and passing the windowpane test.
Shape the dough into a ball, place in an oiled bowl, cover with a kitchen towel, and let rise in a warm spot for 60 to 75 minutes until doubled.
Punch the dough down and turn it onto a lightly floured surface. Divide into 10 to 12 equal pieces, about 75 g each.
Roll each piece into a rope about 20 cm / 8 inches long. Form into a ring, pinching the ends together firmly to seal, and place on a parchment-lined 33 x 46 cm / 13 x 18 inch sheet pan, spaced 5 cm apart.
Cover loosely and proof 20 to 25 minutes until the rings look puffy but haven't doubled.
Heat the oven to 180 C / 356 F while the rings finish proofing.
Brush each ring with the beaten egg and sprinkle with coarse sugar.
Bake 18 to 22 minutes until deep golden brown and the rings sound hollow when tapped on the bottom.
Cool on a wire rack at least 10 minutes before serving.