Jam-Filled Challah Doughnuts

Mix yeast, 1 Tbsp. sugar, and 1/2 cup warm water in the bowl of a stand mixer fitted with paddle attachment to combine; let sit until mixture is foamy, 8–10 minutes.
Add egg yolks, salt, and remaining 1/4 cup sugar and beat on medium-low speed to combine, about 1 minute. Increase speed to medium and gradually add 2 1/3 cups flour; mix until combined and thick, about 3 minutes. Remove paddle, scrape down mixture back into bottom of bowl, and swap in dough hook.
With mixer on low speed, add butter a piece at a time, mixing until incorporated after each addition. Once all the butter has been added, mix until dough is smooth, shiny, and elastic, about 5 minutes.
Turn dough out onto a lightly floured surface and knead, adding more flour by the tablespoonful, until no longer sticky, about 3 minutes. Place dough in a lightly oiled large bowl and turn to coat. Cover loosely with plastic wrap and let rise in a warm, draft-free spot until doubled in size, about 1 hour.
Punch down dough and roll or pat out on a lightly floured surface to 1/2" thick. Punch out rounds of dough with cutter and place on a parchment-lined baking sheet, spacing 1" apart. Reroll scraps once and cut out more rounds. Cover doughnuts loosely with plastic wrap and let rise until not quite doubled in size, 30–40 minutes.
Fit a large heavy saucepan with thermometer; pour in oil to come 3" up sides and heat over medium-high until thermometer registers 350°F. Working in batches and adjusting heat to maintain oil temperature, fry doughnuts, turning once, until golden brown, about 2 minutes. Transfer to a paper towel–lined baking sheet; let cool slightly.
Place sugar in a medium bowl, and working in batches, toss doughnuts to coat. Transfer back to baking sheet.
Stir jam to loosen and spoon into pastry bag fitted with tip (or use a squeeze bottle). Working one at a time, poke a small hole into side of doughnuts (a chopstick works great) and gently pipe in a generous amount of jam.