Haskell, 103 bytes

head.([1..]>>=).g 1 0
g b i a d=[[]|a<1]++[j:o|j<-[max(div(b-1)a)i+1..div(b*d)a],o<-g(b*j)j(a*j-b)$d-1]

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This uses iterative deepening depth-first search, using the bound \$\frac a{bd} ≤ \frac 1j ≤ \frac ab\$ on the first term \$\frac1j\$ of a size-\$d\$ Egyptian fraction for \$\frac ab\$. Unclear if this qualifies as “fast”, but it does solve \$n = 4\$ in a few seconds:

\$4 = \frac11 + \frac12 + \frac13 + \frac14 + \frac15 + \frac16 + \frac17 + \frac18 + \frac19 + \frac1{10} + \frac1{11} + \frac1{12} + \frac1{13} + \frac1{14} + \frac1{15} + \frac1{16} + \frac1{17} + \frac1{18} + \frac1{19} + \frac1{20} + \frac1{21} + \frac1{22} + \frac1{23} + \frac1{24} + \frac1{25} + \frac1{26} + \frac1{27} + \frac1{28} + \frac1{30} + \frac1{34} + \frac1{100} + \frac1{11934} + \frac1{14536368}.\$\begin{split} 4 = {}&\frac11 + \frac12 + \frac13 + \frac14 + \frac15 + \frac16 + \frac17 + \frac18 + \frac19 + \frac1{10} + \frac1{11} + \frac1{12} + \frac1{13} \\ &+ \frac1{14} + \frac1{15} + \frac1{16} + \frac1{17} + \frac1{18} + \frac1{19} + \frac1{20} + \frac1{21} + \frac1{22} + \frac1{23} + \frac1{24} \\ &+ \frac1{25} + \frac1{26} + \frac1{27} + \frac1{28} + \frac1{30} + \frac1{34} + \frac1{100} + \frac1{11934} + \frac1{14536368}. \end{split}